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Magaard, Shaska, Shpectorov, and Völklein list smooth Riemann surfaces of genus \( g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Breuer.
They list a genus 6 Riemann surface with automorphism group (39,1) in the GAP library of small groups. The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (3,3,13).
We use Magma to compute equations of this curve. The main tools are the Eichler trace formula and black-box commands in Magma for obtaining matrix generators of a representation of a finite group having a specified character.
Magma V2.21-7 Tue Mar 22 2016 08:23:07 on Davids-MacBook-Pro-2 [Seed =
2401820926]
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Type ? for help. Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(39,1);
> RunExample(G,6,[3,3,13]);
Set seed to 0.
Character Table of Group G
--------------------------
---------------------------------------
Class | 1 2 3 4 5 6 7
Size | 1 13 13 3 3 3 3
Order | 1 3 3 13 13 13 13
---------------------------------------
p = 3 1 1 1 4 5 6 7
p = 13 1 2 3 1 1 1 1
---------------------------------------
X.1 + 1 1 1 1 1 1 1
X.2 0 1-1-J J 1 1 1 1
X.3 0 1 J-1-J 1 1 1 1
X.4 0 3 0 0 Z1 Z1#2 Z1#4 Z1#7
X.5 0 3 0 0 Z1#7 Z1 Z1#2 Z1#4
X.6 0 3 0 0 Z1#4 Z1#7 Z1 Z1#2
X.7 0 3 0 0 Z1#2 Z1#4 Z1#7 Z1
Explanation of Character Value Symbols
--------------------------------------
# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k
J = RootOfUnity(3)
Z1 = (CyclotomicField(13: Sparse := true)) ! [ RationalField() | 0, 0, 0, 0,
0, 0, 0, 1, 1, 0, 0, 1 ]
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 3 Length 13
Rep G.1^2
[3] Order 3 Length 13
Rep G.1
[4] Order 13 Length 3
Rep G.2^4
[5] Order 13 Length 3
Rep G.2^7
[6] Order 13 Length 3
Rep G.2
[7] Order 13 Length 3
Rep G.2^2
Surface kernel generators: [ G.1 * G.2^5, G.1^2 * G.2^5, G.2^2 ]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 1, 1, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 1, 1, 0 ]
I_2 =[ 0, 0, 0, 1, 0, 0, 1 ]
S_2 =[ 0, 0, 0, 2, 2, 1, 2 ]
H^0(C,2K)=[ 0, 0, 0, 1, 2, 1, 1 ]
I_3 =[ 2, 1, 1, 2, 2, 2, 3 ]
S_3 =[ 3, 1, 1, 4, 4, 4, 5 ]
H^0(C,3K)=[ 1, 0, 0, 2, 2, 2, 2 ]
I2timesS1=[ 2, 2, 2, 3, 2, 3, 2 ]
Is clearly not generated by quadrics? true
Plane quintic obstruction? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 39 and degree 24
[
[z^23 + z^17 - z^12 + z^10 + z^4 + 1 z^14 - z^9 - z^3 + z z^23 - z^18 + z^17
- z^15 - z^12 + z^10 - z^6 + z^4 - 1 0 0 0]
[-z^23 + z^18 - z^17 + z^15 + z^14 + z^12 - z^10 - z^9 + z^6 - z^4 - z^3 + z
+ 1 -z^23 - z^17 + z^12 - z^10 - z^4 - 1 z^23 + z^17 + z^14 - z^12 +
z^10 - z^9 + z^4 - z^3 + z + 1 0 0 0]
[0 1 0 0 0 0]
[0 0 0 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 -
2*z^6 + z^4 - 3*z^3 + 3*z - 1) 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 +
2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1) 1/3*(-z^23 - z^18 - z^17 - z^15 +
z^12 - z^10 - z^6 - z^4 + 1)]
[0 0 0 -1 -z^14 + z^9 + z^3 - z -1]
[0 0 0 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 +
2) 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 -
1) 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 +
1)],
[-z^23 - z^17 - z^14 + z^12 - z^10 + z^9 - z^4 + z^3 - z - 1 -z^14 + z^9 +
z^3 - z -z^23 + z^18 - z^17 + z^15 + z^12 - z^10 + z^6 - z^4 0 0 0]
[z^23 + z^17 - z^12 + z^10 + z^4 1 -z^14 + z^9 + z^3 - z 0 0 0]
[z^14 - z^9 - z^3 + z + 1 -z^23 + z^18 - z^17 + z^15 + z^12 - z^10 + z^6 -
z^4 z^23 + z^17 - z^12 + z^10 + z^4 0 0 0]
[0 0 0 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 -
2*z^6 + z^4 - 3*z^3 + 3*z - 1) 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 +
2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1) 1/3*(-z^23 - z^18 - z^17 - z^15 +
z^12 - z^10 - z^6 - z^4 + 1)]
[0 0 0 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 - 2)
1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 - 2)
1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 - 1)]
[0 0 0 1 0 0]
]
Matrix Surface Kernel Generators:
[
[z^14 - z^9 - z^3 + z -z^23 - z^17 + z^12 - z^10 - z^4 1 0 0 0]
[-z^23 - z^17 + z^12 - z^10 - z^4 - 1 -z^14 + z^9 + z^3 - z - 1 z^14 - z^9 -
z^3 + z 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 - 1)
1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 - 1) 1/3*(-z^23
- z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 + 1)]
[0 0 0 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 -
2*z^6 + z^4 - 3*z^3 + 3*z - 1) 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 +
2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1) 1/3*(-z^23 - z^18 - z^17 - z^15 +
z^12 - z^10 - z^6 - z^4 + 1)]
[0 0 0 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 +
1) 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 + 3*z^14 - 2*z^12 + 2*z^10 - 3*z^9
- z^6 + 2*z^4 - 3*z^3 + 3*z + 1) 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 +
z^10 + z^6 + z^4 + 2)],
[0 1 0 0 0 0]
[-z^23 - z^17 - z^14 + z^12 - z^10 + z^9 - z^4 + z^3 - z - 1 -z^14 + z^9 +
z^3 - z -z^23 + z^18 - z^17 + z^15 + z^12 - z^10 + z^6 - z^4 0 0 0]
[-z^23 - z^17 + z^12 - z^10 - z^4 - 1 -z^14 + z^9 + z^3 - z - 1 z^14 - z^9 -
z^3 + z 0 0 0]
[0 0 0 -1 -z^14 + z^9 + z^3 - z -1]
[0 0 0 0 1 0]
[0 0 0 1 0 0],
[-z^23 + z^18 - z^17 + z^15 + z^14 + z^12 - z^10 - z^9 + z^6 - z^4 - z^3 + z
+ 1 -z^23 - z^17 + z^12 - z^10 - z^4 - 1 z^23 + z^17 + z^14 - z^12 +
z^10 - z^9 + z^4 - z^3 + z + 1 0 0 0]
[-z^14 + z^9 + z^3 - z - 1 z^23 + z^17 - z^12 + z^10 + z^4 -z^23 - z^17 +
z^12 - z^10 - z^4 - 1 0 0 0]
[2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 z^23 + z^17 +
z^14 - z^12 + z^10 - z^9 + z^4 - z^3 + z + 1 -z^14 + z^9 + z^3 - z - 1 0
0 0]
[0 0 0 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 - 1)
1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 + 2)
1/3*(2*z^23 - z^18 + 2*z^17 - z^15 + 3*z^14 - 2*z^12 + 2*z^10 - 3*z^9 -
z^6 + 2*z^4 - 3*z^3 + 3*z - 2)]
[0 0 0 -1 -z^14 + z^9 + z^3 - z -1]
[0 0 0 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 -
2*z^6 + z^4 - 3*z^3 + 3*z - 1) 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 +
2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1) 1/3*(-z^23 - z^18 - z^17 - z^15 +
z^12 - z^10 - z^6 - z^4 + 1)]
]
Finding quadrics:
I2 contains a 3-dimensional subspace of CharacterRow 4
Dimension 6
Multiplicity 2
[
x_0^2 + (z^14 - z^9 - z^3 + z)*x_1^2 + 2*x_1*x_2 + (z^23 + z^17 - z^12 +
z^10 + z^4)*x_2^2,
x_0*x_1 + 1/2*(-z^23 - z^17 + z^12 - z^10 - z^4)*x_1^2 + 1/2*x_2^2,
x_0*x_2 + 1/2*x_1^2 + 1/2*(-z^14 + z^9 + z^3 - z)*x_2^2,
x_3^2 + 1/3*(-2*z^23 + 4*z^18 - 2*z^17 + 4*z^15 - 3*z^14 + 2*z^12 - 2*z^10 +
3*z^9 + 4*z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_4^2 + 1/3*(-z^23 + 5*z^18 -
z^17 + 5*z^15 + z^12 - z^10 + 5*z^6 - z^4 + 4)*x_4*x_5 + 1/3*(z^23 +
z^18 + z^17 + z^15 - 3*z^14 - z^12 + z^10 + 3*z^9 + z^6 + z^4 + 3*z^3 -
3*z - 1)*x_5^2,
x_3*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 +
1)*x_4^2 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 -
2*z^4 + 2)*x_4*x_5 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 +
2*z^10 - z^6 + 2*z^4 + 1)*x_5^2,
x_3*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 -
1)*x_4^2 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 -
z^4 - 2)*x_4*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 +
z^4 - 1)*x_5^2
]
I2 contains a 3-dimensional subspace of CharacterRow 7
Dimension 6
Multiplicity 2
[
x_0^2 + (-z^14 + z^9 + z^3 - z)*x_1^2 + (-z^23 - z^17 - z^14 + z^12 - z^10 +
z^9 - z^4 + z^3 - z - 3)*x_1*x_2 + (-z^23 - z^17 + z^12 - z^10 -
z^4)*x_2^2,
x_0*x_1 + (-z^14 + z^9 + z^3 - z)*x_1*x_2 - x_2^2,
x_0*x_2 - x_1^2 + (-z^23 - z^17 + z^12 - z^10 - z^4)*x_1*x_2,
x_0*x_3 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 -
2*z^4 + 2)*x_1*x_3 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 +
2*z^12 - 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_1*x_4 +
1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 +
1)*x_1*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 -
1)*x_2*x_3 - x_2*x_4 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 -
z^12 + z^10 - 3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z - 1)*x_2*x_5,
x_0*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 -
2*z^4 - 1)*x_1*x_3 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 -
2*z^6 + z^4 - 1)*x_1*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10
- z^6 - z^4 - 2)*x_1*x_5 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12
- 2*z^10 + z^6 - 2*z^4 + 2)*x_2*x_3 + (-z^14 + z^9 + z^3 - z)*x_2*x_4 +
1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 -
1)*x_2*x_5,
x_0*x_5 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 +
1)*x_1*x_3 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 - 3*z^14 + z^12 - z^10
+ 3*z^9 + 2*z^6 - z^4 + 3*z^3 - 3*z + 1)*x_1*x_4 + 1/3*(-2*z^23 + z^18 -
2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1)*x_1*x_5 + 1/3*(2*z^23
- z^18 + 2*z^17 - z^15 + 3*z^14 - 2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4
- 3*z^3 + 3*z - 2)*x_2*x_3 - x_2*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 +
z^12 - z^10 - z^6 - z^4 + 1)*x_2*x_5
]
Finding cubics:
I3 contains a 2-dimensional subspace of CharacterRow 1
Dimension 3
Multiplicity 3
[
x_0^3 + (-z^23 - z^17 + z^12 - z^10 - z^4)*x_0^2*x_1 + (-z^14 + z^9 + z^3 -
z)*x_0^2*x_2 + (-z^14 + z^9 + z^3 - z)*x_0*x_1^2 + (z^23 + z^17 + z^14 -
z^12 + z^10 - z^9 + z^4 - z^3 + z - 1)*x_0*x_1*x_2 + (-z^23 - z^17 +
z^12 - z^10 - z^4)*x_0*x_2^2 + x_1^3 + (z^18 + z^15 + z^6)*x_1^2*x_2 +
(z^23 - z^18 + z^17 - z^15 + z^14 - z^12 + z^10 - z^9 - z^6 + z^4 - z^3
+ z - 1)*x_1*x_2^2 + x_2^3,
x_0^2*x_3 + 1/3*(-z^23 - z^18 - z^17 - z^15 - 2*z^14 + z^12 - z^10 + 2*z^9 -
z^6 - z^4 + 2*z^3 - 2*z - 3)*x_0^2*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 +
z^15 + 2*z^14 + 2*z^12 - 2*z^10 - 2*z^9 + z^6 - 2*z^4 - 2*z^3 +
2*z)*x_0^2*x_5 + 2*x_0*x_1*x_3 + 2*x_0*x_1*x_4 + 1/3*(2*z^23 + 2*z^18 +
2*z^17 + 2*z^15 - 2*z^14 - 2*z^12 + 2*z^10 + 2*z^9 + 2*z^6 + 2*z^4 +
2*z^3 - 2*z)*x_0*x_1*x_5 + 1/3*(2*z^23 + 2*z^18 + 2*z^17 + 2*z^15 -
2*z^14 - 2*z^12 + 2*z^10 + 2*z^9 + 2*z^6 + 2*z^4 + 2*z^3 -
2*z)*x_0*x_2*x_3 + 1/3*(-4*z^23 + 2*z^18 - 4*z^17 + 2*z^15 + 4*z^14 +
4*z^12 - 4*z^10 - 4*z^9 + 2*z^6 - 4*z^4 - 4*z^3 + 4*z)*x_0*x_2*x_4 +
(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 + 2*z^14 - 4*z^12 + 4*z^10 - 2*z^9 -
2*z^6 + 4*z^4 - 2*z^3 + 2*z)*x_0*x_2*x_5 + 1/3*(-2*z^23 + z^18 - 2*z^17
+ z^15 + 2*z^14 + 2*z^12 - 2*z^10 - 2*z^9 + z^6 - 2*z^4 - 2*z^3 +
2*z)*x_1^2*x_3 + 1/3*(-z^23 - z^18 - z^17 - z^15 - 2*z^14 + z^12 - z^10
+ 2*z^9 - z^6 - z^4 + 2*z^3 - 2*z - 3)*x_1^2*x_4 + 1/3*(-2*z^23 + z^18 -
2*z^17 + z^15 + 2*z^14 + 2*z^12 - 2*z^10 - 2*z^9 + z^6 - 2*z^4 - 2*z^3 +
2*z)*x_1^2*x_5 + 2*x_1*x_2*x_3 + 1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15
- 4*z^14 + 2*z^12 - 2*z^10 + 4*z^9 - 2*z^6 - 2*z^4 + 4*z^3 - 4*z -
6)*x_1*x_2*x_4 + 1/3*(-4*z^23 + 2*z^18 - 4*z^17 + 2*z^15 + 4*z^14 +
4*z^12 - 4*z^10 - 4*z^9 + 2*z^6 - 4*z^4 - 4*z^3 + 4*z)*x_1*x_2*x_5 +
1/3*(z^23 + z^18 + z^17 + z^15 - z^14 - z^12 + z^10 + z^9 + z^6 + z^4 +
z^3 - z)*x_2^2*x_3 + x_2^2*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 -
2*z^14 + z^12 - z^10 + 2*z^9 - z^6 - z^4 + 2*z^3 - 2*z - 3)*x_2^2*x_5,
x_3^3 + (-z^23 - z^17 + z^12 - z^10 - z^4)*x_3^2*x_4 + (z^18 + z^15 +
z^6)*x_3^2*x_5 + (-z^14 + z^9 + z^3 - z)*x_3*x_4^2 + (-z^23 + z^18 -
z^17 + z^15 + z^12 - z^10 + z^6 - z^4)*x_3*x_4*x_5 + (z^23 - z^18 + z^17
- z^15 + z^14 - z^12 + z^10 - z^9 - z^6 + z^4 - z^3 + z - 1)*x_3*x_5^2 +
x_4^3 + (-z^14 + z^9 + z^3 - z)*x_4^2*x_5 + (-z^23 - z^17 + z^12 - z^10
- z^4)*x_4*x_5^2 + x_5^3
]
I3 contains a 1-dimensional subspace of CharacterRow 2
Dimension 1
Multiplicity 1
[
x_0^2*x_3 + 1/3*(-2*z^23 - z^22 - z^20 - z^19 + z^18 - 2*z^17 - z^16 + z^15
- 2*z^13 + 2*z^12 - z^11 - 2*z^10 - z^8 + z^5 - 2*z^4 + z^2 - z +
1)*x_0^2*x_4 + 1/3*(-z^23 + z^22 - 2*z^20 + z^19 + 2*z^18 - z^17 + z^16
+ 2*z^15 - 3*z^14 + 2*z^13 + z^12 - 2*z^11 - z^10 + 3*z^9 - 2*z^8 +
3*z^6 - z^5 - z^4 + 3*z^3 - z^2 - 2*z + 2)*x_0^2*x_5 +
2*z^13*x_0*x_1*x_3 + (-2*z^13 - 2)*x_0*x_1*x_4 + 1/3*(4*z^23 - 4*z^22 +
2*z^20 + 2*z^19 - 2*z^18 + 4*z^17 - 4*z^16 - 2*z^15 + 6*z^14 - 2*z^13 -
4*z^12 + 2*z^11 + 4*z^10 - 6*z^9 + 2*z^8 - 2*z^5 + 4*z^4 - 6*z^3 - 2*z^2
+ 2*z - 2)*x_0*x_1*x_5 + 1/3*(-2*z^23 + 2*z^22 - 4*z^20 + 2*z^19 -
2*z^18 - 2*z^17 + 2*z^16 - 2*z^15 - 6*z^14 + 4*z^13 + 2*z^12 - 4*z^11 -
2*z^10 + 6*z^9 - 4*z^8 - 2*z^5 - 2*z^4 + 6*z^3 - 2*z^2 - 4*z +
4)*x_0*x_2*x_3 + 1/3*(4*z^23 - 4*z^22 + 2*z^20 - 4*z^19 - 2*z^18 +
4*z^17 - 4*z^16 - 2*z^15 + 6*z^14 - 2*z^13 - 4*z^12 + 2*z^11 + 4*z^10 -
6*z^9 + 2*z^8 - 6*z^6 + 4*z^5 + 4*z^4 - 6*z^3 + 4*z^2 + 2*z -
2)*x_0*x_2*x_4 + 1/3*(2*z^23 - 2*z^22 + z^20 - 2*z^19 - z^18 + 2*z^17 -
2*z^16 - z^15 + 3*z^14 - z^13 - 2*z^12 + z^11 + 2*z^10 - 3*z^9 + z^8 -
3*z^6 + 2*z^5 + 2*z^4 - 3*z^3 + 2*z^2 + z - 1)*x_1^2*x_3 + 1/3*(z^23 +
2*z^22 - z^20 + 2*z^19 - 2*z^18 + z^17 + 2*z^16 - 2*z^15 + z^13 - z^12 -
z^11 + z^10 - z^8 - 2*z^5 + z^4 - 2*z^2 + 2*z - 2)*x_1^2*x_4 +
1/3*(-z^23 + z^22 + z^20 + z^19 - z^18 - z^17 + z^16 - z^15 - z^13 +
z^12 + z^11 - z^10 + z^8 - z^5 - z^4 - z^2 + z - 1)*x_1^2*x_5 +
2*x_1*x_2*x_3 + 1/3*(-4*z^23 - 2*z^22 - 2*z^20 - 2*z^19 + 2*z^18 -
4*z^17 - 2*z^16 + 2*z^15 - 4*z^13 + 4*z^12 - 2*z^11 - 4*z^10 - 2*z^8 +
2*z^5 - 4*z^4 + 2*z^2 - 2*z + 2)*x_1*x_2*x_4 + 1/3*(-2*z^23 + 2*z^22 -
4*z^20 + 2*z^19 + 4*z^18 - 2*z^17 + 2*z^16 + 4*z^15 - 6*z^14 + 4*z^13 +
2*z^12 - 4*z^11 - 2*z^10 + 6*z^9 - 4*z^8 + 6*z^6 - 2*z^5 - 2*z^4 + 6*z^3
- 2*z^2 - 4*z + 4)*x_1*x_2*x_5 + 1/3*(-z^23 + z^22 + z^20 - 2*z^19 +
2*z^18 - z^17 + z^16 + 2*z^15 - z^13 + z^12 + z^11 - z^10 + z^8 + 2*z^5
- z^4 + 2*z^2 + z - 1)*x_2^2*x_3 + z^13*x_2^2*x_4 + 1/3*(z^23 + 2*z^22 -
z^20 + 2*z^19 - 2*z^18 + z^17 + 2*z^16 - 2*z^15 + z^13 - z^12 - z^11 +
z^10 - z^8 - 2*z^5 + z^4 - 2*z^2 + 2*z - 2)*x_2^2*x_5
]
I3 contains a 1-dimensional subspace of CharacterRow 3
Dimension 1
Multiplicity 1
[
x_0^2*x_3 + 1/3*(-z^23 + z^22 + z^20 + z^19 - z^18 - z^17 + z^16 - z^15 +
2*z^13 + z^12 + z^11 - z^10 + z^8 - z^5 - z^4 - z^2 + z + 2)*x_0^2*x_4 +
1/3*(z^23 - z^22 + 2*z^20 - z^19 + z^18 + z^17 - z^16 + z^15 - 2*z^13 -
z^12 + 2*z^11 + z^10 + 2*z^8 + z^5 + z^4 + z^2 - z - 2)*x_0^2*x_5 +
(-2*z^13 - 2)*x_0*x_1*x_3 + 2*z^13*x_0*x_1*x_4 + 1/3*(2*z^23 + 4*z^22 -
2*z^20 - 2*z^19 + 2*z^18 + 2*z^17 + 4*z^16 + 2*z^15 + 2*z^13 - 2*z^12 -
2*z^11 + 2*z^10 - 2*z^8 + 2*z^5 + 2*z^4 + 2*z^2 + 4*z + 2)*x_0*x_1*x_5 +
1/3*(2*z^23 - 2*z^22 + 4*z^20 - 2*z^19 - 4*z^18 + 2*z^17 - 2*z^16 -
4*z^15 - 4*z^13 - 2*z^12 + 4*z^11 + 2*z^10 + 4*z^8 - 6*z^6 + 2*z^5 +
2*z^4 + 2*z^2 - 2*z - 4)*x_0*x_2*x_3 + 1/3*(2*z^23 + 4*z^22 - 2*z^20 +
4*z^19 - 4*z^18 + 2*z^17 + 4*z^16 - 4*z^15 + 2*z^13 - 2*z^12 - 2*z^11 +
2*z^10 - 2*z^8 - 4*z^5 + 2*z^4 - 4*z^2 + 4*z + 2)*x_0*x_2*x_4 +
1/3*(z^23 + 2*z^22 - z^20 + 2*z^19 - 2*z^18 + z^17 + 2*z^16 - 2*z^15 +
z^13 - z^12 - z^11 + z^10 - z^8 - 2*z^5 + z^4 - 2*z^2 + 2*z +
1)*x_1^2*x_3 + 1/3*(2*z^23 - 2*z^22 + z^20 - 2*z^19 - z^18 + 2*z^17 -
2*z^16 - z^15 + 3*z^14 - z^13 - 2*z^12 + z^11 + 2*z^10 - 3*z^9 + z^8 -
3*z^6 + 2*z^5 + 2*z^4 - 3*z^3 + 2*z^2 + z - 4)*x_1^2*x_4 + 1/3*(-2*z^23
- z^22 - z^20 - z^19 + z^18 - 2*z^17 - z^16 + z^15 + z^13 + 2*z^12 -
z^11 - 2*z^10 - z^8 + z^5 - 2*z^4 + z^2 - z + 1)*x_1^2*x_5 +
2*x_1*x_2*x_3 + 1/3*(-2*z^23 + 2*z^22 + 2*z^20 + 2*z^19 - 2*z^18 -
2*z^17 + 2*z^16 - 2*z^15 + 4*z^13 + 2*z^12 + 2*z^11 - 2*z^10 + 2*z^8 -
2*z^5 - 2*z^4 - 2*z^2 + 2*z + 4)*x_1*x_2*x_4 + 1/3*(2*z^23 - 2*z^22 +
4*z^20 - 2*z^19 + 2*z^18 + 2*z^17 - 2*z^16 + 2*z^15 - 4*z^13 - 2*z^12 +
4*z^11 + 2*z^10 + 4*z^8 + 2*z^5 + 2*z^4 + 2*z^2 - 2*z - 4)*x_1*x_2*x_5 +
1/3*(-2*z^23 - z^22 - z^20 + 2*z^19 + z^18 - 2*z^17 - z^16 + z^15 + z^13
+ 2*z^12 - z^11 - 2*z^10 - z^8 + 3*z^6 - 2*z^5 - 2*z^4 - 2*z^2 - z +
1)*x_2^2*x_3 + (-z^13 - 1)*x_2^2*x_4 + 1/3*(2*z^23 - 2*z^22 + z^20 -
2*z^19 - z^18 + 2*z^17 - 2*z^16 - z^15 + 3*z^14 - z^13 - 2*z^12 + z^11 +
2*z^10 - 3*z^9 + z^8 - 3*z^6 + 2*z^5 + 2*z^4 - 3*z^3 + 2*z^2 + z -
4)*x_2^2*x_5
]
I3 contains a 6-dimensional subspace of CharacterRow 4
Dimension 12
Multiplicity 4
[
x_0^3 + 1/3*(z^23 + 2*z^18 + z^17 + 2*z^15 + 2*z^14 - z^12 + z^10 - 2*z^9 +
2*z^6 + z^4 - 2*z^3 + 2*z + 5)*x_0*x_1^2 + 1/3*(2*z^23 + 2*z^17 + 2*z^14
- 2*z^12 + 2*z^10 - 2*z^9 + 2*z^4 - 2*z^3 + 2*z - 2)*x_0*x_1*x_2 +
1/3*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 + 3*z^14 - 4*z^12 + 4*z^10 -
3*z^9 - 2*z^6 + 4*z^4 - 3*z^3 + 3*z + 3)*x_0*x_2^2 + 1/3*(-2*z^23 - z^18
- 2*z^17 - z^15 - 2*z^14 + 2*z^12 - 2*z^10 + 2*z^9 - z^6 - 2*z^4 + 2*z^3
- 2*z - 4)*x_1^3 + 1/3*(-5*z^23 - 5*z^17 - 10*z^14 + 5*z^12 - 5*z^10 +
10*z^9 - 5*z^4 + 10*z^3 - 10*z - 9)*x_1^2*x_2 + 1/3*(-10*z^23 - 10*z^17
- 5*z^14 + 10*z^12 - 10*z^10 + 5*z^9 - 10*z^4 + 5*z^3 - 5*z -
9)*x_1*x_2^2 + 1/3*(-3*z^23 + z^18 - 3*z^17 + z^15 - 3*z^14 + 3*z^12 -
3*z^10 + 3*z^9 + z^6 - 3*z^4 + 3*z^3 - 3*z - 3)*x_2^3,
x_0^2*x_1 + 1/3*(-2*z^23 - 2*z^17 + z^14 + 2*z^12 - 2*z^10 - z^9 - 2*z^4 -
z^3 + z - 1)*x_0*x_1^2 + 1/3*(-2*z^23 - 2*z^17 - 2*z^14 + 2*z^12 -
2*z^10 + 2*z^9 - 2*z^4 + 2*z^3 - 2*z + 2)*x_0*x_1*x_2 + 1/3*(-z^23 +
2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4)*x_0*x_2^2 +
1/3*(z^23 + z^17 - z^14 - z^12 + z^10 + z^9 + z^4 + z^3 - z - 1)*x_1^3 +
1/3*(4*z^23 - 4*z^18 + 4*z^17 - 4*z^15 + z^14 - 4*z^12 + 4*z^10 - z^9 -
4*z^6 + 4*z^4 - z^3 + z - 2)*x_1^2*x_2 + 1/3*(-z^23 + z^18 - z^17 + z^15
- 4*z^14 + z^12 - z^10 + 4*z^9 + z^6 - z^4 + 4*z^3 - 4*z + 2)*x_1*x_2^2
+ 1/3*(z^23 + z^17 - z^12 + z^10 + z^4 - 1)*x_2^3,
x_0^2*x_2 + 1/3*(2*z^23 - 2*z^18 + 2*z^17 - 2*z^15 + z^14 - 2*z^12 + 2*z^10
- z^9 - 2*z^6 + 2*z^4 - z^3 + z - 2)*x_0*x_1^2 + 1/3*(-2*z^23 - 2*z^17 -
2*z^14 + 2*z^12 - 2*z^10 + 2*z^9 - 2*z^4 + 2*z^3 - 2*z + 2)*x_0*x_1*x_2
+ 1/3*(z^23 + z^17 - 2*z^14 - z^12 + z^10 + 2*z^9 + z^4 + 2*z^3 - 2*z -
1)*x_0*x_2^2 + 1/3*(z^14 - z^9 - z^3 + z - 1)*x_1^3 + 1/3*(-3*z^23 -
z^18 - 3*z^17 - z^15 + 3*z^12 - 3*z^10 - z^6 - 3*z^4 + 1)*x_1^2*x_2 +
1/3*(-3*z^23 + 4*z^18 - 3*z^17 + 4*z^15 + 3*z^12 - 3*z^10 + 4*z^6 -
3*z^4 + 2)*x_1*x_2^2 + 1/3*(-z^23 - z^17 + z^14 + z^12 - z^10 - z^9 -
z^4 - z^3 + z - 1)*x_2^3,
x_0^2*x_3 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 -
2*z^4 + 2)*x_0*x_1*x_3 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 -
z^6 - z^4 + 1)*x_0*x_1*x_4 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12
+ 2*z^10 - z^6 + 2*z^4 - 2)*x_0*x_2*x_3 + 1/3*(-z^23 - z^18 - z^17 -
z^15 + z^12 - z^10 - z^6 - z^4 + 1)*x_0*x_2*x_5 + 1/3*(-2*z^23 + z^18 -
2*z^17 + z^15 - 3*z^14 + 2*z^12 - 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 -
3*z + 2)*x_1^2*x_3 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6
+ z^4 - 1)*x_1^2*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 +
z^6 + z^4 - 4)*x_1*x_2*x_3 + x_1*x_2*x_4 + 1/3*(z^23 - 2*z^18 + z^17 -
2*z^15 - z^12 + z^10 - 2*z^6 + z^4 - 1)*x_1*x_2*x_5 + 1/3*(-z^23 - z^18
- z^17 - z^15 + z^12 - z^10 - z^6 - z^4 - 2)*x_2^2*x_3 + 1/3*(z^23 +
z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 - 1)*x_2^2*x_4,
x_0^2*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 -
2*z^4 + 2)*x_0*x_1*x_3 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 -
z^6 - z^4 - 2)*x_0*x_1*x_4 - x_0*x_1*x_5 + 1/3*(-z^23 + 2*z^18 - z^17 +
2*z^15 + z^12 - z^10 + 2*z^6 - z^4 + 1)*x_0*x_2*x_3 + (-z^14 + z^9 + z^3
- z)*x_0*x_2*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 -
z^4 - 2)*x_0*x_2*x_5 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10
- 2*z^6 + z^4 - 1)*x_1^2*x_3 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 +
z^10 + z^6 + z^4 + 2)*x_1^2*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 -
3*z^14 - z^12 + z^10 + 3*z^9 + z^6 + z^4 + 3*z^3 - 3*z + 2)*x_1*x_2*x_3
+ (z^14 - z^9 - z^3 + z)*x_1*x_2*x_4 + 1/3*(4*z^23 - 2*z^18 + 4*z^17 -
2*z^15 + 3*z^14 - 4*z^12 + 4*z^10 - 3*z^9 - 2*z^6 + 4*z^4 - 3*z^3 + 3*z
- 1)*x_1*x_2*x_5 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10
- z^6 + 2*z^4 - 2)*x_2^2*x_3 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 +
2*z^12 - 2*z^10 + z^6 - 2*z^4 + 2)*x_2^2*x_4 + x_2^2*x_5,
x_0^2*x_5 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4
+ 1)*x_0*x_1*x_3 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 + 2*z^12
- 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_0*x_1*x_4 + (-z^23 -
z^17 + z^12 - z^10 - z^4)*x_0*x_1*x_5 + 1/3*(z^23 - 2*z^18 + z^17 -
2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z -
1)*x_0*x_2*x_3 - x_0*x_2*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 -
3*z^14 + 2*z^12 - 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z +
2)*x_0*x_2*x_5 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 - 3*z^14 + z^12 -
z^10 + 3*z^9 + 2*z^6 - z^4 + 3*z^3 - 3*z + 1)*x_1^2*x_3 + x_1^2*x_4 +
1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 -
2)*x_1^2*x_5 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 -
z^6 + 2*z^4 - 5)*x_1*x_2*x_3 + (z^18 + z^15 + z^6 + 1)*x_1*x_2*x_4 +
1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 +
1)*x_1*x_2*x_5 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 -
2*z^6 + z^4 - 1)*x_2^2*x_3 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 + 3*z^14
- 2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 + 3*z - 2)*x_2^2*x_4,
x_0*x_3^2 + 1/117*(-28*z^23 - z^18 - 28*z^17 - z^15 + 3*z^14 + 28*z^12 -
28*z^10 - 3*z^9 - z^6 - 28*z^4 - 3*z^3 + 3*z - 98)*x_1*x_3^2 +
1/117*(22*z^23 + 76*z^18 + 22*z^17 + 76*z^15 + 6*z^14 - 22*z^12 +
22*z^10 - 6*z^9 + 76*z^6 + 22*z^4 - 6*z^3 + 6*z - 40)*x_1*x_3*x_4 +
1/117*(74*z^23 - 106*z^18 + 74*z^17 - 106*z^15 + 84*z^14 - 74*z^12 +
74*z^10 - 84*z^9 - 106*z^6 + 74*z^4 - 84*z^3 + 84*z - 92)*x_1*x_3*x_5 +
1/39*(-5*z^23 + 4*z^18 - 5*z^17 + 4*z^15 - 12*z^14 + 5*z^12 - 5*z^10 +
12*z^9 + 4*z^6 - 5*z^4 + 12*z^3 - 12*z + 2)*x_1*x_4^2 + 1/117*(-56*z^23
- 2*z^18 - 56*z^17 - 2*z^15 + 6*z^14 + 56*z^12 - 56*z^10 - 6*z^9 - 2*z^6
- 56*z^4 - 6*z^3 + 6*z + 38)*x_1*x_4*x_5 + 1/39*(-5*z^23 + 4*z^18 -
5*z^17 + 4*z^15 - 12*z^14 + 5*z^12 - 5*z^10 + 12*z^9 + 4*z^6 - 5*z^4 +
12*z^3 - 12*z + 41)*x_1*x_5^2 + 1/117*(-50*z^23 + 40*z^18 - 50*z^17 +
40*z^15 - 120*z^14 + 50*z^12 - 50*z^10 + 120*z^9 + 40*z^6 - 50*z^4 +
120*z^3 - 120*z + 59)*x_2*x_3^2 + 1/39*(10*z^23 - 8*z^18 + 10*z^17 -
8*z^15 + 24*z^14 - 10*z^12 + 10*z^10 - 24*z^9 - 8*z^6 + 10*z^4 - 24*z^3
+ 24*z - 4)*x_2*x_3*x_4 + 1/117*(56*z^23 + 2*z^18 + 56*z^17 + 2*z^15 -
6*z^14 - 56*z^12 + 56*z^10 + 6*z^9 + 2*z^6 + 56*z^4 + 6*z^3 - 6*z -
38)*x_2*x_3*x_5 + 1/39*(5*z^23 - 4*z^18 + 5*z^17 - 4*z^15 + 12*z^14 -
5*z^12 + 5*z^10 - 12*z^9 - 4*z^6 + 5*z^4 - 12*z^3 + 12*z - 41)*x_2*x_4^2
+ 1/39*(10*z^23 - 8*z^18 + 10*z^17 - 8*z^15 + 24*z^14 - 10*z^12 +
10*z^10 - 24*z^9 - 8*z^6 + 10*z^4 - 24*z^3 + 24*z - 4)*x_2*x_4*x_5 +
1/117*(28*z^23 + z^18 + 28*z^17 + z^15 - 3*z^14 - 28*z^12 + 28*z^10 +
3*z^9 + z^6 + 28*z^4 + 3*z^3 - 3*z - 19)*x_2*x_5^2,
x_0*x_3*x_4 + 1/117*(23*z^23 + 5*z^18 + 23*z^17 + 5*z^15 - 15*z^14 - 23*z^12
+ 23*z^10 + 15*z^9 + 5*z^6 + 23*z^4 + 15*z^3 - 15*z + 22)*x_1*x_3^2 +
1/117*(-149*z^23 + 49*z^18 - 149*z^17 + 49*z^15 - 30*z^14 + 149*z^12 -
149*z^10 + 30*z^9 + 49*z^6 - 149*z^4 + 30*z^3 - 30*z + 5)*x_1*x_3*x_4 +
1/117*(59*z^23 + 23*z^18 + 59*z^17 + 23*z^15 + 48*z^14 - 59*z^12 +
59*z^10 - 48*z^9 + 23*z^6 + 59*z^4 - 48*z^3 + 48*z + 31)*x_1*x_3*x_5 +
1/13*(4*z^23 + 2*z^18 + 4*z^17 + 2*z^15 - 6*z^14 - 4*z^12 + 4*z^10 +
6*z^9 + 2*z^6 + 4*z^4 + 6*z^3 - 6*z + 1)*x_1*x_4^2 + 1/117*(-71*z^23 +
10*z^18 - 71*z^17 + 10*z^15 - 30*z^14 + 71*z^12 - 71*z^10 + 30*z^9 +
10*z^6 - 71*z^4 + 30*z^3 - 30*z - 73)*x_1*x_4*x_5 + 1/39*(-z^23 - 7*z^18
- z^17 - 7*z^15 + 21*z^14 + z^12 - z^10 - 21*z^9 - 7*z^6 - z^4 - 21*z^3
+ 21*z + 16)*x_1*x_5^2 + 1/117*(55*z^23 - 44*z^18 + 55*z^17 - 44*z^15 +
15*z^14 - 55*z^12 + 55*z^10 - 15*z^9 - 44*z^6 + 55*z^4 - 15*z^3 + 15*z +
17)*x_2*x_3^2 + 1/39*(-37*z^23 + 14*z^18 - 37*z^17 + 14*z^15 - 3*z^14 +
37*z^12 - 37*z^10 + 3*z^9 + 14*z^6 - 37*z^4 + 3*z^3 - 3*z +
7)*x_2*x_3*x_4 + 1/117*(71*z^23 - 10*z^18 + 71*z^17 - 10*z^15 + 30*z^14
- 71*z^12 + 71*z^10 - 30*z^9 - 10*z^6 + 71*z^4 - 30*z^3 + 30*z +
73)*x_2*x_3*x_5 + 1/39*(z^23 + 7*z^18 + z^17 + 7*z^15 - 21*z^14 - z^12 +
z^10 + 21*z^9 + 7*z^6 + z^4 + 21*z^3 - 21*z - 16)*x_2*x_4^2 +
1/13*(-8*z^23 + 9*z^18 - 8*z^17 + 9*z^15 - z^14 + 8*z^12 - 8*z^10 + z^9
+ 9*z^6 - 8*z^4 + z^3 - z - 2)*x_2*x_4*x_5 + 1/117*(16*z^23 + 34*z^18 +
16*z^17 + 34*z^15 + 15*z^14 - 16*z^12 + 16*z^10 - 15*z^9 + 34*z^6 +
16*z^4 - 15*z^3 + 15*z + 56)*x_2*x_5^2,
x_0*x_3*x_5 + 1/117*(70*z^23 - 56*z^18 + 70*z^17 - 56*z^15 + 51*z^14 -
70*z^12 + 70*z^10 - 51*z^9 - 56*z^6 + 70*z^4 - 51*z^3 + 51*z -
106)*x_1*x_3^2 + 1/117*(23*z^23 + 5*z^18 + 23*z^17 + 5*z^15 + 102*z^14 -
23*z^12 + 23*z^10 - 102*z^9 + 5*z^6 + 23*z^4 - 102*z^3 + 102*z -
95)*x_1*x_3*x_4 + 1/117*(10*z^23 - 125*z^18 + 10*z^17 - 125*z^15 +
24*z^14 - 10*z^12 + 10*z^10 - 24*z^9 - 125*z^6 + 10*z^4 - 24*z^3 + 24*z
+ 35)*x_1*x_3*x_5 + 1/13*(2*z^23 + z^18 + 2*z^17 + z^15 - 3*z^14 -
2*z^12 + 2*z^10 + 3*z^9 + z^6 + 2*z^4 + 3*z^3 - 3*z - 6)*x_1*x_4^2 +
1/117*(-55*z^23 + 44*z^18 - 55*z^17 + 44*z^15 - 15*z^14 + 55*z^12 -
55*z^10 + 15*z^9 + 44*z^6 - 55*z^4 + 15*z^3 - 15*z - 17)*x_1*x_4*x_5 +
1/39*(-7*z^23 + 29*z^18 - 7*z^17 + 29*z^15 - 9*z^14 + 7*z^12 - 7*z^10 +
9*z^9 + 29*z^6 - 7*z^4 + 9*z^3 - 9*z + 34)*x_1*x_5^2 + 1/117*(8*z^23 +
17*z^18 + 8*z^17 + 17*z^15 - 51*z^14 - 8*z^12 + 8*z^10 + 51*z^9 + 17*z^6
+ 8*z^4 + 51*z^3 - 51*z + 28)*x_2*x_3^2 + 1/13*(-4*z^23 - 2*z^18 -
4*z^17 - 2*z^15 + 6*z^14 + 4*z^12 - 4*z^10 - 6*z^9 - 2*z^6 - 4*z^4 -
6*z^3 + 6*z + 12)*x_2*x_3*x_4 + 1/117*(16*z^23 + 34*z^18 + 16*z^17 +
34*z^15 - 102*z^14 - 16*z^12 + 16*z^10 + 102*z^9 + 34*z^6 + 16*z^4 +
102*z^3 - 102*z + 56)*x_2*x_3*x_5 + 1/39*(7*z^23 - 29*z^18 + 7*z^17 -
29*z^15 + 9*z^14 - 7*z^12 + 7*z^10 - 9*z^9 - 29*z^6 + 7*z^4 - 9*z^3 +
9*z - 34)*x_2*x_4^2 + 1/13*(-4*z^23 - 2*z^18 - 4*z^17 - 2*z^15 + 6*z^14
+ 4*z^12 - 4*z^10 - 6*z^9 - 2*z^6 - 4*z^4 - 6*z^3 + 6*z - 1)*x_2*x_4*x_5
+ 1/117*(86*z^23 - 22*z^18 + 86*z^17 - 22*z^15 + 66*z^14 - 86*z^12 +
86*z^10 - 66*z^9 - 22*z^6 + 86*z^4 - 66*z^3 + 66*z - 50)*x_2*x_5^2,
x_0*x_4^2 + 1/117*(2*z^23 - 25*z^18 + 2*z^17 - 25*z^15 + 75*z^14 - 2*z^12 +
2*z^10 - 75*z^9 - 25*z^6 + 2*z^4 - 75*z^3 + 75*z + 7)*x_1*x_3^2 +
1/117*(4*z^23 - 50*z^18 + 4*z^17 - 50*z^15 - 84*z^14 - 4*z^12 + 4*z^10 +
84*z^9 - 50*z^6 + 4*z^4 + 84*z^3 - 84*z - 220)*x_1*x_3*x_4 +
1/117*(-100*z^23 + 80*z^18 - 100*z^17 + 80*z^15 - 6*z^14 + 100*z^12 -
100*z^10 + 6*z^9 + 80*z^6 - 100*z^4 + 6*z^3 - 6*z + 118)*x_1*x_3*x_5 +
1/39*(5*z^23 - 4*z^18 + 5*z^17 - 4*z^15 + 12*z^14 - 5*z^12 + 5*z^10 -
12*z^9 - 4*z^6 + 5*z^4 - 12*z^3 + 12*z + 37)*x_1*x_4^2 + 1/117*(4*z^23 -
50*z^18 + 4*z^17 - 50*z^15 - 84*z^14 - 4*z^12 + 4*z^10 + 84*z^9 - 50*z^6
+ 4*z^4 + 84*z^3 - 84*z + 14)*x_1*x_4*x_5 + 1/13*(-7*z^23 + 3*z^18 -
7*z^17 + 3*z^15 + 4*z^14 + 7*z^12 - 7*z^10 - 4*z^9 + 3*z^6 - 7*z^4 -
4*z^3 + 4*z - 5)*x_1*x_5^2 + 1/117*(-2*z^23 + 25*z^18 - 2*z^17 + 25*z^15
+ 42*z^14 + 2*z^12 - 2*z^10 - 42*z^9 + 25*z^6 - 2*z^4 - 42*z^3 + 42*z +
110)*x_2*x_3^2 + 1/13*(-12*z^23 - 6*z^18 - 12*z^17 - 6*z^15 - 8*z^14 +
12*z^12 - 12*z^10 + 8*z^9 - 6*z^6 - 12*z^4 + 8*z^3 - 8*z -
16)*x_2*x_3*x_4 + 1/117*(-4*z^23 + 50*z^18 - 4*z^17 + 50*z^15 + 84*z^14
+ 4*z^12 - 4*z^10 - 84*z^9 + 50*z^6 - 4*z^4 - 84*z^3 + 84*z -
14)*x_2*x_3*x_5 + 1/13*(7*z^23 - 3*z^18 + 7*z^17 - 3*z^15 - 4*z^14 -
7*z^12 + 7*z^10 + 4*z^9 - 3*z^6 + 7*z^4 + 4*z^3 - 4*z + 5)*x_2*x_4^2 +
1/39*(-10*z^23 + 8*z^18 - 10*z^17 + 8*z^15 - 24*z^14 + 10*z^12 - 10*z^10
+ 24*z^9 + 8*z^6 - 10*z^4 + 24*z^3 - 24*z + 4)*x_2*x_4*x_5 +
1/117*(-41*z^23 - 14*z^18 - 41*z^17 - 14*z^15 + 42*z^14 + 41*z^12 -
41*z^10 - 42*z^9 - 14*z^6 - 41*z^4 - 42*z^3 + 42*z + 32)*x_2*x_5^2,
x_0*x_4*x_5 + 1/117*(z^23 + 46*z^18 + z^17 + 46*z^15 - 21*z^14 - z^12 + z^10
+ 21*z^9 + 46*z^6 + z^4 + 21*z^3 - 21*z + 62)*x_1*x_3^2 +
1/117*(-76*z^23 + 14*z^18 - 76*z^17 + 14*z^15 - 42*z^14 + 76*z^12 -
76*z^10 + 42*z^9 + 14*z^6 - 76*z^4 + 42*z^3 - 42*z - 32)*x_1*x_3*x_4 +
1/117*(106*z^23 - 38*z^18 + 106*z^17 - 38*z^15 + 114*z^14 - 106*z^12 +
106*z^10 - 114*z^9 - 38*z^6 + 106*z^4 - 114*z^3 + 114*z -
97)*x_1*x_3*x_5 + 1/39*(22*z^23 - 2*z^18 + 22*z^17 - 2*z^15 + 6*z^14 -
22*z^12 + 22*z^10 - 6*z^9 - 2*z^6 + 22*z^4 - 6*z^3 + 6*z - 1)*x_1*x_4^2
+ 1/117*(-76*z^23 + 14*z^18 - 76*z^17 + 14*z^15 - 42*z^14 + 76*z^12 -
76*z^10 + 42*z^9 + 14*z^6 - 76*z^4 + 42*z^3 - 42*z - 32)*x_1*x_4*x_5 +
1/39*(-17*z^23 - 2*z^18 - 17*z^17 - 2*z^15 + 6*z^14 + 17*z^12 - 17*z^10
- 6*z^9 - 2*z^6 - 17*z^4 - 6*z^3 + 6*z + 38)*x_1*x_5^2 + 1/117*(38*z^23
- 7*z^18 + 38*z^17 - 7*z^15 + 21*z^14 - 38*z^12 + 38*z^10 - 21*z^9 -
7*z^6 + 38*z^4 - 21*z^3 + 21*z + 16)*x_2*x_3^2 + 1/39*(-31*z^23 +
17*z^18 - 31*z^17 + 17*z^15 - 12*z^14 + 31*z^12 - 31*z^10 + 12*z^9 +
17*z^6 - 31*z^4 + 12*z^3 - 12*z - 11)*x_2*x_3*x_4 + 1/117*(37*z^23 -
53*z^18 + 37*z^17 - 53*z^15 + 42*z^14 - 37*z^12 + 37*z^10 - 42*z^9 -
53*z^6 + 37*z^4 - 42*z^3 + 42*z + 71)*x_2*x_3*x_5 + 1/39*(17*z^23 +
2*z^18 + 17*z^17 + 2*z^15 - 6*z^14 - 17*z^12 + 17*z^10 + 6*z^9 + 2*z^6 +
17*z^4 + 6*z^3 - 6*z + 1)*x_2*x_4^2 + 1/39*(-5*z^23 + 4*z^18 - 5*z^17 +
4*z^15 - 12*z^14 + 5*z^12 - 5*z^10 + 12*z^9 + 4*z^6 - 5*z^4 + 12*z^3 -
12*z - 37)*x_2*x_4*x_5 + 1/117*(-z^23 + 71*z^18 - z^17 + 71*z^15 +
21*z^14 + z^12 - z^10 - 21*z^9 + 71*z^6 - z^4 - 21*z^3 + 21*z +
55)*x_2*x_5^2,
x_0*x_5^2 + 1/117*(59*z^23 - 94*z^18 + 59*z^17 - 94*z^15 + 48*z^14 - 59*z^12
+ 59*z^10 - 48*z^9 - 94*z^6 + 59*z^4 - 48*z^3 + 48*z - 86)*x_1*x_3^2 +
1/117*(40*z^23 - 32*z^18 + 40*z^17 - 32*z^15 + 96*z^14 - 40*z^12 +
40*z^10 - 96*z^9 - 32*z^6 + 40*z^4 - 96*z^3 + 96*z - 94)*x_1*x_3*x_4 +
1/117*(14*z^23 - 58*z^18 + 14*z^17 - 58*z^15 - 60*z^14 - 14*z^12 +
14*z^10 + 60*z^9 - 58*z^6 + 14*z^4 + 60*z^3 - 60*z + 166)*x_1*x_3*x_5 +
1/39*(11*z^23 - z^18 + 11*z^17 - z^15 + 3*z^14 - 11*z^12 + 11*z^10 -
3*z^9 - z^6 + 11*z^4 - 3*z^3 + 3*z - 20)*x_1*x_4^2 + 1/117*(40*z^23 -
32*z^18 + 40*z^17 - 32*z^15 + 96*z^14 - 40*z^12 + 40*z^10 - 96*z^9 -
32*z^6 + 40*z^4 - 96*z^3 + 96*z - 94)*x_1*x_4*x_5 + 1/39*(-2*z^23 +
25*z^18 - 2*z^17 + 25*z^15 + 3*z^14 + 2*z^12 - 2*z^10 - 3*z^9 + 25*z^6 -
2*z^4 - 3*z^3 + 3*z - 7)*x_1*x_5^2 + 1/117*(-20*z^23 + 16*z^18 - 20*z^17
+ 16*z^15 - 48*z^14 + 20*z^12 - 20*z^10 + 48*z^9 + 16*z^6 - 20*z^4 +
48*z^3 - 48*z + 47)*x_2*x_3^2 + 1/39*(-22*z^23 + 2*z^18 - 22*z^17 +
2*z^15 - 6*z^14 + 22*z^12 - 22*z^10 + 6*z^9 + 2*z^6 - 22*z^4 + 6*z^3 -
6*z + 40)*x_2*x_3*x_4 + 1/117*(38*z^23 + 110*z^18 + 38*z^17 + 110*z^15 -
96*z^14 - 38*z^12 + 38*z^10 + 96*z^9 + 110*z^6 + 38*z^4 + 96*z^3 - 96*z
+ 16)*x_2*x_3*x_5 + 1/39*(2*z^23 - 25*z^18 + 2*z^17 - 25*z^15 - 3*z^14 -
2*z^12 + 2*z^10 + 3*z^9 - 25*z^6 + 2*z^4 + 3*z^3 - 3*z + 7)*x_2*x_4^2 +
1/39*(-22*z^23 + 2*z^18 - 22*z^17 + 2*z^15 - 6*z^14 + 22*z^12 - 22*z^10
+ 6*z^9 + 2*z^6 - 22*z^4 + 6*z^3 - 6*z + 40)*x_2*x_4*x_5 +
1/117*(97*z^23 - 101*z^18 + 97*z^17 - 101*z^15 + 69*z^14 - 97*z^12 +
97*z^10 - 69*z^9 - 101*z^6 + 97*z^4 - 69*z^3 + 69*z - 70)*x_2*x_5^2
]
I3 contains a 6-dimensional subspace of CharacterRow 5
Dimension 12
Multiplicity 4
[
x_0^3 + (3*z^14 - 3*z^9 - 3*z^3 + 3*z)*x_0*x_1^2 + 6*x_0*x_1*x_2 + (3*z^23 +
3*z^17 - 3*z^12 + 3*z^10 + 3*z^4)*x_0*x_2^2 + (-z^23 - z^17 - z^14 +
z^12 - z^10 + z^9 - z^4 + z^3 - z - 1)*x_1^3 + (-z^23 - z^17 - z^14 +
z^12 - z^10 + z^9 - z^4 + z^3 - z - 1)*x_2^3,
x_0^2*x_1 + (-z^23 - z^17 + z^12 - z^10 - z^4)*x_0*x_1^2 + x_0*x_2^2 +
1/3*(z^23 - z^18 + z^17 - z^15 - z^12 + z^10 - z^6 + z^4 - 1)*x_1^3 +
x_1^2*x_2 + 1/3*(-z^14 + z^9 + z^3 - z)*x_2^3,
x_0^2*x_2 + x_0*x_1^2 + (-z^14 + z^9 + z^3 - z)*x_0*x_2^2 + 1/3*(-z^23 -
z^17 + z^12 - z^10 - z^4)*x_1^3 + x_1*x_2^2 + 1/3*(-z^23 + z^18 - z^17 +
z^15 + z^12 - z^10 + z^6 - z^4)*x_2^3,
x_0^2*x_3 + 1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15 + 2*z^12 - 2*z^10 -
2*z^6 - 2*z^4 - 4)*x_0*x_1*x_3 + 1/3*(2*z^23 + 2*z^18 + 2*z^17 + 2*z^15
- 2*z^12 + 2*z^10 + 2*z^6 + 2*z^4 - 2)*x_0*x_1*x_4 + 1/3*(-4*z^23 +
2*z^18 - 4*z^17 + 2*z^15 - 6*z^14 + 4*z^12 - 4*z^10 + 6*z^9 + 2*z^6 -
4*z^4 + 6*z^3 - 6*z + 4)*x_0*x_2*x_3 + 1/3*(2*z^23 + 2*z^18 + 2*z^17 +
2*z^15 - 2*z^12 + 2*z^10 + 2*z^6 + 2*z^4 - 2)*x_0*x_2*x_5 + 1/3*(2*z^23
- z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 + 1)*x_1^2*x_3 +
1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 +
1)*x_1^2*x_4 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4
- 1)*x_1^2*x_5 + 2*x_1*x_2*x_3 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 +
z^12 - z^10 + 2*z^6 - z^4 + 1)*x_2^2*x_3 + 1/3*(z^23 + z^18 + z^17 +
z^15 - z^12 + z^10 + z^6 + z^4 - 1)*x_2^2*x_4 - x_2^2*x_5,
x_0^2*x_4 + 1/3*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 - 4*z^12 + 4*z^10 - 2*z^6
+ 4*z^4 - 4)*x_0*x_1*x_3 + 1/3*(-4*z^23 + 2*z^18 - 4*z^17 + 2*z^15 +
4*z^12 - 4*z^10 + 2*z^6 - 4*z^4 + 4)*x_0*x_1*x_4 + 2*x_0*x_1*x_5 +
1/3*(2*z^23 - 4*z^18 + 2*z^17 - 4*z^15 - 2*z^12 + 2*z^10 - 4*z^6 + 2*z^4
- 2)*x_0*x_2*x_3 + 1/3*(2*z^23 + 2*z^18 + 2*z^17 + 2*z^15 - 2*z^12 +
2*z^10 + 2*z^6 + 2*z^4 + 4)*x_0*x_2*x_5 + 1/3*(2*z^23 - z^18 + 2*z^17 -
z^15 + 3*z^14 - 2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 + 3*z +
1)*x_1^2*x_3 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4
- 2)*x_1^2*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 +
z^6 - 2*z^4 + 2)*x_1^2*x_5 + 2*x_1*x_2*x_4 + 1/3*(-z^23 - z^18 - z^17 -
z^15 + z^12 - z^10 - z^6 - z^4 - 2)*x_2^2*x_3 + 1/3*(z^23 + z^18 + z^17
+ z^15 - z^12 + z^10 + z^6 + z^4 + 2)*x_2^2*x_4 + (-z^14 + z^9 + z^3 -
z)*x_2^2*x_5,
x_0^2*x_5 + 1/3*(2*z^23 - 4*z^18 + 2*z^17 - 4*z^15 - 2*z^12 + 2*z^10 - 4*z^6
+ 2*z^4 - 2)*x_0*x_1*x_3 + 1/3*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 +
6*z^14 - 4*z^12 + 4*z^10 - 6*z^9 - 2*z^6 + 4*z^4 - 6*z^3 + 6*z -
4)*x_0*x_1*x_4 + 1/3*(-2*z^23 + 4*z^18 - 2*z^17 + 4*z^15 - 6*z^14 +
2*z^12 - 2*z^10 + 6*z^9 + 4*z^6 - 2*z^4 + 6*z^3 - 6*z + 2)*x_0*x_2*x_3 +
2*x_0*x_2*x_4 + 1/3*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 - 4*z^12 + 4*z^10
- 2*z^6 + 4*z^4 - 4)*x_0*x_2*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12
+ z^10 + z^6 + z^4 + 2)*x_1^2*x_3 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15
+ 2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1)*x_1^2*x_4 + 1/3*(2*z^23 - z^18 +
2*z^17 - z^15 + 3*z^14 - 2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 +
3*z - 2)*x_1^2*x_5 + 2*x_1*x_2*x_5 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15
+ 2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1)*x_2^2*x_3 + 1/3*(2*z^23 - z^18 +
2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 - 2)*x_2^2*x_4 -
x_2^2*x_5,
x_0*x_3^2 + 1/3*(-2*z^23 + 4*z^18 - 2*z^17 + 4*z^15 - 3*z^14 + 2*z^12 -
2*z^10 + 3*z^9 + 4*z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_0*x_4^2 + 1/3*(-z^23
+ 5*z^18 - z^17 + 5*z^15 + z^12 - z^10 + 5*z^6 - z^4 + 4)*x_0*x_4*x_5 +
1/3*(z^23 + z^18 + z^17 + z^15 - 3*z^14 - z^12 + z^10 + 3*z^9 + z^6 +
z^4 + 3*z^3 - 3*z - 1)*x_0*x_5^2 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 -
z^12 + z^10 - 2*z^6 + z^4 - 4)*x_1*x_3^2 + 1/3*(z^23 + z^18 + z^17 +
z^15 - z^12 + z^10 + z^6 + z^4 + 5)*x_1*x_3*x_4 + (-z^23 - z^17 - z^14 +
z^12 - z^10 + z^9 - z^4 + z^3 - z + 2)*x_1*x_3*x_5 + 1/3*(-2*z^23 + z^18
- 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 + 2)*x_1*x_4^2 + (z^18 +
z^15 - z^14 + z^9 + z^6 + z^3 - z)*x_1*x_4*x_5 + (z^18 + z^15 +
z^6)*x_1*x_5^2 + 1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15 - 3*z^14 +
2*z^12 - 2*z^10 + 3*z^9 - 2*z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_2*x_3^2 +
1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 +
1)*x_2*x_3*x_4 + 1/3*(-2*z^23 + 7*z^18 - 2*z^17 + 7*z^15 - 3*z^14 +
2*z^12 - 2*z^10 + 3*z^9 + 7*z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_2*x_3*x_5 +
1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 +
4)*x_2*x_4^2 + (z^23 + z^17 - z^14 - z^12 + z^10 + z^9 + z^4 + z^3 - z -
1)*x_2*x_4*x_5 + 1/3*(-z^23 - z^18 - z^17 - z^15 + 3*z^14 + z^12 - z^10
- 3*z^9 - z^6 - z^4 - 3*z^3 + 3*z - 5)*x_2*x_5^2,
x_0*x_3*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 +
1)*x_0*x_4^2 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 +
z^6 - 2*z^4 + 2)*x_0*x_4*x_5 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 -
2*z^12 + 2*z^10 - z^6 + 2*z^4 + 1)*x_0*x_5^2 + 1/3*(2*z^23 - z^18 +
2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 + 1)*x_1*x_3^2 +
1/3*(-z^23 - z^18 - z^17 - z^15 + 3*z^14 + z^12 - z^10 - 3*z^9 - z^6 -
z^4 - 3*z^3 + 3*z + 1)*x_1*x_3*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 +
z^12 - z^10 - z^6 - z^4 - 2)*x_1*x_4^2 + x_1*x_4*x_5 + 1/3*(2*z^23 -
z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 - 2)*x_2*x_3^2 +
1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 +
2)*x_2*x_3*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 -
z^4 + 1)*x_2*x_3*x_5 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 -
2*z^10 + z^6 - 2*z^4 + 2)*x_2*x_4^2 + (z^23 + z^17 - z^12 + z^10 + z^4 +
1)*x_2*x_4*x_5 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 + 2*z^12 -
2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z - 1)*x_2*x_5^2,
x_0*x_3*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 -
1)*x_0*x_4^2 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6
- z^4 - 2)*x_0*x_4*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 +
z^6 + z^4 - 1)*x_0*x_5^2 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 +
z^10 - 2*z^6 + z^4 - 1)*x_1*x_3^2 + 1/3*(z^23 + z^18 + z^17 + z^15 -
z^12 + z^10 + z^6 + z^4 + 2)*x_1*x_3*x_4 + (-z^23 + z^18 - z^17 + z^15 -
z^14 + z^12 - z^10 + z^9 + z^6 - z^4 + z^3 - z + 1)*x_1*x_3*x_5 +
1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 +
2)*x_1*x_4^2 + (z^23 + z^17 - z^12 + z^10 + z^4)*x_1*x_4*x_5 - x_1*x_5^2
+ 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 + 2*z^12 - 2*z^10 + 3*z^9
+ z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_2*x_3^2 + 1/3*(2*z^23 - z^18 + 2*z^17
- z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 + 1)*x_2*x_3*x_4 + 1/3*(z^23 +
z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 - 4)*x_2*x_3*x_5 +
1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 +
1)*x_2*x_4^2 + (z^23 + z^17 - z^12 + z^10 + z^4)*x_2*x_4*x_5 +
1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 -
2)*x_2*x_5^2,
x_3^3 + 1/3*(4*z^18 + 4*z^15 + 2*z^14 - 2*z^9 + 4*z^6 - 2*z^3 + 2*z -
3)*x_3*x_4^2 + 1/3*(8*z^23 + 8*z^17 + 4*z^14 - 8*z^12 + 8*z^10 - 4*z^9 +
8*z^4 - 4*z^3 + 4*z - 4)*x_3*x_4*x_5 + 1/3*(-3*z^23 + 3*z^18 - 3*z^17 +
3*z^15 - 4*z^14 + 3*z^12 - 3*z^10 + 4*z^9 + 3*z^6 - 3*z^4 + 4*z^3 - 4*z
- 4)*x_3*x_5^2 + 1/3*(z^18 + z^15 - 2*z^14 + 2*z^9 + z^6 + 2*z^3 - 2*z -
3)*x_4^3 + 1/3*(-3*z^23 - 3*z^18 - 3*z^17 - 3*z^15 + 8*z^14 + 3*z^12 -
3*z^10 - 8*z^9 - 3*z^6 - 3*z^4 - 8*z^3 + 8*z - 4)*x_4^2*x_5 +
1/3*(9*z^23 - 7*z^18 + 9*z^17 - 7*z^15 + 4*z^14 - 9*z^12 + 9*z^10 -
4*z^9 - 7*z^6 + 9*z^4 - 4*z^3 + 4*z + 3)*x_4*x_5^2 + 1/3*(-2*z^18 -
2*z^15 - 2*z^6 - 4)*x_5^3,
x_3^2*x_4 + 1/9*(-10*z^23 + 2*z^18 - 10*z^17 + 2*z^15 - 3*z^14 + 10*z^12 -
10*z^10 + 3*z^9 + 2*z^6 - 10*z^4 + 3*z^3 - 3*z + 4)*x_3*x_4^2 +
1/9*(-2*z^23 + 10*z^18 - 2*z^17 + 10*z^15 + 2*z^12 - 2*z^10 + 10*z^6 -
2*z^4 + 14)*x_3*x_4*x_5 + 1/9*(2*z^23 - z^18 + 2*z^17 - z^15 - 6*z^14 -
2*z^12 + 2*z^10 + 6*z^9 - z^6 + 2*z^4 + 6*z^3 - 6*z - 2)*x_3*x_5^2 +
1/9*(2*z^23 - z^18 + 2*z^17 - z^15 - 3*z^14 - 2*z^12 + 2*z^10 + 3*z^9 -
z^6 + 2*z^4 + 3*z^3 - 3*z + 1)*x_4^3 + 1/9*(-10*z^23 + 5*z^18 - 10*z^17
+ 5*z^15 - 6*z^14 + 10*z^12 - 10*z^10 + 6*z^9 + 5*z^6 - 10*z^4 + 6*z^3 -
6*z - 8)*x_4^2*x_5 + 1/9*(4*z^23 + z^18 + 4*z^17 + z^15 + 12*z^14 -
4*z^12 + 4*z^10 - 12*z^9 + z^6 + 4*z^4 - 12*z^3 + 12*z + 2)*x_4*x_5^2 +
1/9*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 - 4*z^12 + 4*z^10 - 2*z^6 + 4*z^4
- 1)*x_5^3,
x_3^2*x_5 + 1/9*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 + 6*z^14 - 4*z^12 +
4*z^10 - 6*z^9 - 2*z^6 + 4*z^4 - 6*z^3 + 6*z - 7)*x_3*x_4^2 +
1/9*(2*z^23 - 4*z^18 + 2*z^17 - 4*z^15 + 6*z^14 - 2*z^12 + 2*z^10 -
6*z^9 - 4*z^6 + 2*z^4 - 6*z^3 + 6*z - 8)*x_3*x_4*x_5 + 1/9*(z^23 +
4*z^18 + z^17 + 4*z^15 - z^12 + z^10 + 4*z^6 + z^4 - 7)*x_3*x_5^2 +
1/9*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 - 4)*x_4^3 +
1/9*(-5*z^23 - 2*z^18 - 5*z^17 - 2*z^15 + 5*z^12 - 5*z^10 - 2*z^6 -
5*z^4 - 1)*x_4^2*x_5 + 1/9*(2*z^23 - z^18 + 2*z^17 - z^15 + 3*z^14 -
2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 + 3*z + 10)*x_4*x_5^2 +
1/9*(2*z^23 - 4*z^18 + 2*z^17 - 4*z^15 - 2*z^12 + 2*z^10 - 4*z^6 + 2*z^4
- 2)*x_5^3
]
I3 contains a 6-dimensional subspace of CharacterRow 6
Dimension 12
Multiplicity 4
[
x_0^3 + 1/3*(-2*z^23 - z^18 - 2*z^17 - z^15 - 4*z^14 + 2*z^12 - 2*z^10 +
4*z^9 - z^6 - 2*z^4 + 4*z^3 - 4*z - 4)*x_0*x_1^2 + 1/3*(-10*z^23 -
10*z^17 - 10*z^14 + 10*z^12 - 10*z^10 + 10*z^9 - 10*z^4 + 10*z^3 - 10*z
- 14)*x_0*x_1*x_2 + 1/3*(-5*z^23 + z^18 - 5*z^17 + z^15 - 3*z^14 +
5*z^12 - 5*z^10 + 3*z^9 + z^6 - 5*z^4 + 3*z^3 - 3*z - 3)*x_0*x_2^2 +
1/3*(3*z^23 + z^18 + 3*z^17 + z^15 + 3*z^14 - 3*z^12 + 3*z^10 - 3*z^9 +
z^6 + 3*z^4 - 3*z^3 + 3*z + 3)*x_1^3 + 1/3*(11*z^23 - 4*z^18 + 11*z^17 -
4*z^15 + 10*z^14 - 11*z^12 + 11*z^10 - 10*z^9 - 4*z^6 + 11*z^4 - 10*z^3
+ 10*z + 11)*x_1^2*x_2 + 1/3*(6*z^23 + 4*z^18 + 6*z^17 + 4*z^15 + 7*z^14
- 6*z^12 + 6*z^10 - 7*z^9 + 4*z^6 + 6*z^4 - 7*z^3 + 7*z + 15)*x_1*x_2^2
+ 1/3*(4*z^23 - z^18 + 4*z^17 - z^15 + 4*z^14 - 4*z^12 + 4*z^10 - 4*z^9
- z^6 + 4*z^4 - 4*z^3 + 4*z + 2)*x_2^3,
x_0^2*x_1 + 1/3*(-z^23 - z^17 - z^14 + z^12 - z^10 + z^9 - z^4 + z^3 - z +
1)*x_0*x_1^2 + 1/3*(2*z^23 + 2*z^17 - 4*z^14 - 2*z^12 + 2*z^10 + 4*z^9 +
2*z^4 + 4*z^3 - 4*z - 2)*x_0*x_1*x_2 + 1/3*(z^23 - 2*z^18 + z^17 -
2*z^15 - z^12 + z^10 - 2*z^6 + z^4 - 3)*x_0*x_2^2 + 1/3*(-z^23 - z^17 +
z^12 - z^10 - z^4 + 1)*x_1^3 + 1/3*(-2*z^23 + 4*z^18 - 2*z^17 + 4*z^15 +
z^14 + 2*z^12 - 2*z^10 - z^9 + 4*z^6 - 2*z^4 - z^3 + z + 3)*x_1^2*x_2 +
1/3*(-z^23 - z^17 + 4*z^14 + z^12 - z^10 - 4*z^9 - z^4 - 4*z^3 + 4*z -
2)*x_1*x_2^2 + 1/3*(-z^23 - z^17 + z^14 + z^12 - z^10 - z^9 - z^4 - z^3
+ z + 1)*x_2^3,
x_0^2*x_2 + 1/3*(-2*z^23 + 2*z^18 - 2*z^17 + 2*z^15 - z^14 + 2*z^12 - 2*z^10
+ z^9 + 2*z^6 - 2*z^4 + z^3 - z - 1)*x_0*x_1^2 + 1/3*(-4*z^23 - 4*z^17 +
2*z^14 + 4*z^12 - 4*z^10 - 2*z^9 - 4*z^4 - 2*z^3 + 2*z - 2)*x_0*x_1*x_2
+ 1/3*(-z^23 - z^17 - z^14 + z^12 - z^10 + z^9 - z^4 + z^3 - z +
1)*x_0*x_2^2 + 1/3*(z^23 + z^17 - z^14 - z^12 + z^10 + z^9 + z^4 + z^3 -
z + 1)*x_1^3 + 1/3*(4*z^23 + 4*z^17 - z^14 - 4*z^12 + 4*z^10 + z^9 +
4*z^4 + z^3 - z - 2)*x_1^2*x_2 + 1/3*(5*z^23 - 4*z^18 + 5*z^17 - 4*z^15
+ 2*z^14 - 5*z^12 + 5*z^10 - 2*z^9 - 4*z^6 + 5*z^4 - 2*z^3 + 2*z -
1)*x_1*x_2^2 + 1/3*(-z^14 + z^9 + z^3 - z + 1)*x_2^3,
x_0^2*x_3 + (-z^23 - z^17 + z^12 - z^10 - z^4)*x_0*x_1*x_3 + 1/3*(-z^23 +
2*z^18 - z^17 + 2*z^15 - 3*z^14 + z^12 - z^10 + 3*z^9 + 2*z^6 - z^4 +
3*z^3 - 3*z + 1)*x_0*x_1*x_4 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 +
z^12 - z^10 + 2*z^6 - z^4 + 1)*x_0*x_1*x_5 + 1/3*(-z^23 + 2*z^18 - z^17
+ 2*z^15 - 3*z^14 + z^12 - z^10 + 3*z^9 + 2*z^6 - z^4 + 3*z^3 - 3*z +
1)*x_0*x_2*x_3 - x_0*x_2*x_4 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 +
3*z^14 - 2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 + 3*z -
2)*x_0*x_2*x_5 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 -
2*z^6 + z^4 - 1)*x_1^2*x_3 + x_1^2*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 +
z^15 - 3*z^14 + 2*z^12 - 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z +
2)*x_1^2*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4
+ 2)*x_1*x_2*x_3 + (z^18 + z^15 + z^6)*x_1*x_2*x_4 + 1/3*(z^23 + z^18 +
z^17 + z^15 - z^12 + z^10 + z^6 + z^4 - 4)*x_1*x_2*x_5 + 1/3*(z^23 -
2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 - 2*z^6 + z^4 -
3*z^3 + 3*z - 1)*x_2^2*x_4 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 +
z^10 - 2*z^6 + z^4 - 1)*x_2^2*x_5,
x_0^2*x_4 - x_0*x_1*x_3 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 -
z^6 - z^4 + 1)*x_0*x_1*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 -
z^10 - z^6 - z^4 + 1)*x_0*x_1*x_5 + 1/3*(-z^23 - z^18 - z^17 - z^15 +
z^12 - z^10 - z^6 - z^4 + 1)*x_0*x_2*x_3 + (-z^14 + z^9 + z^3 -
z)*x_0*x_2*x_4 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 +
2*z^6 - z^4 + 1)*x_0*x_2*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 +
z^10 + z^6 + z^4 - 1)*x_1^2*x_3 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 -
z^12 + z^10 - 2*z^6 + z^4 - 1)*x_1^2*x_5 + 1/3*(z^23 - 2*z^18 + z^17 -
2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z -
1)*x_1*x_2*x_3 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 + 2*z^12 -
2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_1*x_2*x_5 + x_2^2*x_3
+ 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 -
1)*x_2^2*x_4 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4
- 1)*x_2^2*x_5,
x_0^2*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 -
1)*x_0*x_1*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 +
z^6 - 2*z^4 - 1)*x_0*x_1*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 +
z^10 + z^6 + z^4 - 1)*x_0*x_2*x_3 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 -
z^12 + z^10 - 2*z^6 + z^4 - 1)*x_0*x_2*x_5 + 1/3*(-z^23 - z^18 - z^17 -
z^15 + z^12 - z^10 - z^6 - z^4 + 1)*x_1^2*x_3 + 1/3*(-z^23 + 2*z^18 -
z^17 + 2*z^15 - 3*z^14 + z^12 - z^10 + 3*z^9 + 2*z^6 - z^4 + 3*z^3 - 3*z
+ 1)*x_1^2*x_5 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 +
2*z^6 - z^4 + 1)*x_1*x_2*x_3 - x_1*x_2*x_4 + 1/3*(2*z^23 - z^18 + 2*z^17
- z^15 + 3*z^14 - 2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 + 3*z -
5)*x_1*x_2*x_5 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 -
z^4 + 1)*x_2^2*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6
- z^4 + 1)*x_2^2*x_5,
x_0*x_3^2 + 1/3*(-2*z^23 + 4*z^18 - 2*z^17 + 4*z^15 - 3*z^14 + 2*z^12 -
2*z^10 + 3*z^9 + 4*z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_0*x_4^2 + 1/3*(-z^23
+ 5*z^18 - z^17 + 5*z^15 + z^12 - z^10 + 5*z^6 - z^4 + 4)*x_0*x_4*x_5 +
1/3*(z^23 + z^18 + z^17 + z^15 - 3*z^14 - z^12 + z^10 + 3*z^9 + z^6 +
z^4 + 3*z^3 - 3*z - 1)*x_0*x_5^2 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 +
z^12 - z^10 + 2*z^6 - z^4 + 4)*x_1*x_3^2 + (-z^23 + z^18 - z^17 + z^15 +
z^12 - z^10 + z^6 - z^4 - 1)*x_1*x_3*x_4 + 1/3*(4*z^23 + z^18 + 4*z^17 +
z^15 + 3*z^14 - 4*z^12 + 4*z^10 - 3*z^9 + z^6 + 4*z^4 - 3*z^3 + 3*z -
4)*x_1*x_3*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - 3*z^14 - z^12 + z^10 +
3*z^9 + z^6 + z^4 + 3*z^3 - 3*z - 1)*x_1*x_4^2 + 1/3*(-4*z^23 - z^18 -
4*z^17 - z^15 + 3*z^14 + 4*z^12 - 4*z^10 - 3*z^9 - z^6 - 4*z^4 - 3*z^3 +
3*z - 5)*x_1*x_4*x_5 + (z^23 - z^18 + z^17 - z^15 + z^14 - z^12 + z^10 -
z^9 - z^6 + z^4 - z^3 + z + 1)*x_1*x_5^2 + 1/3*(4*z^23 - 2*z^18 + 4*z^17
- 2*z^15 + 3*z^14 - 4*z^12 + 4*z^10 - 3*z^9 - 2*z^6 + 4*z^4 - 3*z^3 +
3*z - 4)*x_2*x_3^2 + 1/3*(-z^23 - z^18 - z^17 - z^15 + 3*z^14 + z^12 -
z^10 - 3*z^9 - z^6 - z^4 - 3*z^3 + 3*z - 2)*x_2*x_3*x_4 + 1/3*(z^23 -
5*z^18 + z^17 - 5*z^15 - z^12 + z^10 - 5*z^6 + z^4 + 5)*x_2*x_3*x_5 +
1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 -
4)*x_2*x_4^2 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 +
z^6 - 2*z^4 + 5)*x_2*x_4*x_5 + 1/3*(-2*z^23 + 4*z^18 - 2*z^17 + 4*z^15 -
3*z^14 + 2*z^12 - 2*z^10 + 3*z^9 + 4*z^6 - 2*z^4 + 3*z^3 - 3*z +
2)*x_2*x_5^2,
x_0*x_3*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 +
1)*x_0*x_4^2 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 +
z^6 - 2*z^4 + 2)*x_0*x_4*x_5 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 -
2*z^12 + 2*z^10 - z^6 + 2*z^4 + 1)*x_0*x_5^2 + 1/3*(-2*z^23 + z^18 -
2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 + 2)*x_1*x_3^2 + (-z^23 -
z^17 - z^14 + z^12 - z^10 + z^9 - z^4 + z^3 - z)*x_1*x_3*x_4 +
1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 +
1)*x_1*x_3*x_5 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 -
z^6 + 2*z^4 + 1)*x_1*x_4^2 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 -
3*z^14 + 2*z^12 - 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z -
4)*x_1*x_4*x_5 + (z^14 - z^9 - z^3 + z)*x_1*x_5^2 + 1/3*(-z^23 + 2*z^18
- z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 + 1)*x_2*x_3^2 +
1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 + 2*z^12 - 2*z^10 + 3*z^9 +
z^6 - 2*z^4 + 3*z^3 - 3*z - 1)*x_2*x_3*x_4 + 1/3*(2*z^23 - z^18 + 2*z^17
- z^15 + 3*z^14 - 2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 + 3*z -
2)*x_2*x_3*x_5 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 -
z^6 + 2*z^4 + 1)*x_2*x_4^2 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 -
z^10 - z^6 - z^4 - 2)*x_2*x_4*x_5 + 1/3*(-z^23 - z^18 - z^17 - z^15 +
z^12 - z^10 - z^6 - z^4 + 1)*x_2*x_5^2,
x_0*x_3*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 -
1)*x_0*x_4^2 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6
- z^4 - 2)*x_0*x_4*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 +
z^6 + z^4 - 1)*x_0*x_5^2 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 -
z^10 + 2*z^6 - z^4 + 1)*x_1*x_3^2 - x_1*x_3*x_4 + 1/3*(z^23 - 2*z^18 +
z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z
- 4)*x_1*x_3*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 +
z^4 - 1)*x_1*x_4^2 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6
- z^4 - 2)*x_1*x_4*x_5 + x_1*x_5^2 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15
+ 3*z^14 - z^12 + z^10 - 3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z -
1)*x_2*x_3^2 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4
- 2)*x_2*x_3*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 + 2*z^12
- 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z + 5)*x_2*x_3*x_5 +
1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 -
1)*x_2*x_4^2 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 +
z^6 - 2*z^4 + 2)*x_2*x_4*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 +
z^10 + z^6 + z^4 - 1)*x_2*x_5^2,
x_3^3 + (z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 - 3*z^9 -
2*z^6 + z^4 - 3*z^3 + 3*z - 1)*x_3*x_4^2 + (2*z^23 - 4*z^18 + 2*z^17 -
4*z^15 - 2*z^12 + 2*z^10 - 4*z^6 + 2*z^4 - 2)*x_3*x_4*x_5 + (-2*z^23 +
z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 + 2)*x_3*x_5^2 +
1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 - 2)*x_4^3 +
3*x_4^2*x_5 + (-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 + 2*z^12 - 2*z^10
+ 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_4*x_5^2 + 1/3*(-z^23 + 2*z^18
- z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 + 1)*x_5^3,
x_3^2*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 -
2*z^4 - 1)*x_3*x_4^2 + 1/3*(2*z^23 + 2*z^18 + 2*z^17 + 2*z^15 - 2*z^12 +
2*z^10 + 2*z^6 + 2*z^4 - 2)*x_3*x_4*x_5 + 1/3*(-2*z^23 + z^18 - 2*z^17 +
z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1)*x_3*x_5^2 + 1/9*(2*z^23 - z^18
+ 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 - 2)*x_4^3 + 1/3*(z^23 -
2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 - 1)*x_4*x_5^2 +
1/9*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 - 2)*x_5^3,
x_3^2*x_5 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 +
1)*x_3*x_4^2 + 1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15 + 2*z^12 - 2*z^10
- 2*z^6 - 2*z^4 + 2)*x_3*x_4*x_5 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 +
z^12 - z^10 + 2*z^6 - z^4 + 1)*x_3*x_5^2 + 1/9*(z^23 - 2*z^18 + z^17 -
2*z^15 - z^12 + z^10 - 2*z^6 + z^4 - 1)*x_4^3 + 1/3*(-z^23 + 2*z^18 -
z^17 + 2*z^15 - 3*z^14 + z^12 - z^10 + 3*z^9 + 2*z^6 - z^4 + 3*z^3 - 3*z
+ 1)*x_4*x_5^2 + 1/9*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 +
z^6 - 2*z^4 - 1)*x_5^3
]
I3 contains a 9-dimensional subspace of CharacterRow 7
Dimension 15
Multiplicity 5
[
x_0^2*x_3 + 1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15 - 6*z^14 + 2*z^12 -
2*z^10 + 6*z^9 - 2*z^6 - 2*z^4 + 6*z^3 - 6*z + 2)*x_0*x_2*x_3 +
2*x_0*x_2*x_4 + 1/3*(-2*z^23 + 4*z^18 - 2*z^17 + 4*z^15 - 6*z^14 +
2*z^12 - 2*z^10 + 6*z^9 + 4*z^6 - 2*z^4 + 6*z^3 - 6*z + 2)*x_0*x_2*x_5 +
1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 - 3*z^14 + z^12 - z^10 + 3*z^9 +
2*z^6 - z^4 + 3*z^3 - 3*z + 1)*x_1^2*x_3 + 1/3*(z^23 + z^18 + z^17 +
z^15 - z^12 + z^10 + z^6 + z^4 + 2)*x_1^2*x_4 - x_1^2*x_5 + 1/3*(2*z^23
- 4*z^18 + 2*z^17 - 4*z^15 - 2*z^12 + 2*z^10 - 4*z^6 + 2*z^4 -
2)*x_1*x_2*x_4 + 1/3*(-4*z^23 + 2*z^18 - 4*z^17 + 2*z^15 + 4*z^12 -
4*z^10 + 2*z^6 - 4*z^4 + 4)*x_1*x_2*x_5 + 1/3*(-4*z^23 + 2*z^18 - 4*z^17
+ 2*z^15 + 4*z^12 - 4*z^10 + 2*z^6 - 4*z^4 + 4)*x_2^2*x_3 + (-z^23 -
z^17 - 2*z^14 + z^12 - z^10 + 2*z^9 - z^4 + 2*z^3 - 2*z + 1)*x_2^2*x_4 +
1/3*(-4*z^23 + 5*z^18 - 4*z^17 + 5*z^15 + 4*z^12 - 4*z^10 + 5*z^6 -
4*z^4 + 4)*x_2^2*x_5,
x_0^2*x_4 + 1/3*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 - 4*z^12 + 4*z^10 - 2*z^6
+ 4*z^4 - 4)*x_0*x_2*x_3 + 1/3*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 -
4*z^12 + 4*z^10 - 2*z^6 + 4*z^4 + 2)*x_0*x_2*x_5 + 1/3*(2*z^23 - z^18 +
2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 + 1)*x_1^2*x_3 +
1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 +
2)*x_1^2*x_4 + 1/3*(2*z^23 + 2*z^18 + 2*z^17 + 2*z^15 - 2*z^12 + 2*z^10
+ 2*z^6 + 2*z^4 - 2)*x_1*x_2*x_4 + 1/3*(-4*z^23 + 2*z^18 - 4*z^17 +
2*z^15 + 4*z^12 - 4*z^10 + 2*z^6 - 4*z^4 - 2)*x_1*x_2*x_5 + 1/3*(-4*z^23
+ 2*z^18 - 4*z^17 + 2*z^15 + 4*z^12 - 4*z^10 + 2*z^6 - 4*z^4 -
2)*x_2^2*x_3 + (z^23 - z^18 + z^17 - z^15 - z^12 + z^10 - z^6 + z^4 -
1)*x_2^2*x_4 + 1/3*(-4*z^23 - z^18 - 4*z^17 - z^15 + 4*z^12 - 4*z^10 -
z^6 - 4*z^4 - 5)*x_2^2*x_5,
x_0^2*x_5 + 1/3*(-4*z^23 + 2*z^18 - 4*z^17 + 2*z^15 - 6*z^14 + 4*z^12 -
4*z^10 + 6*z^9 + 2*z^6 - 4*z^4 + 6*z^3 - 6*z + 4)*x_0*x_2*x_3 +
2*x_0*x_2*x_4 + 1/3*(2*z^23 + 2*z^18 + 2*z^17 + 2*z^15 - 6*z^14 - 2*z^12
+ 2*z^10 + 6*z^9 + 2*z^6 + 2*z^4 + 6*z^3 - 6*z - 2)*x_0*x_2*x_5 +
1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 - 1)*x_1^2*x_3
+ 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 +
1)*x_1^2*x_4 + 2*x_1*x_2*x_3 + 1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15 +
2*z^12 - 2*z^10 - 2*z^6 - 2*z^4 + 2)*x_1*x_2*x_4 + 1/3*(-2*z^23 + 4*z^18
- 2*z^17 + 4*z^15 + 2*z^12 - 2*z^10 + 4*z^6 - 2*z^4 + 2)*x_1*x_2*x_5 +
1/3*(-2*z^23 + 4*z^18 - 2*z^17 + 4*z^15 + 2*z^12 - 2*z^10 + 4*z^6 -
2*z^4 + 5)*x_2^2*x_3 + (-z^23 + z^18 - z^17 + z^15 - 2*z^14 + z^12 -
z^10 + 2*z^9 + z^6 - z^4 + 2*z^3 - 2*z + 1)*x_2^2*x_4 + 1/3*(-5*z^23 +
4*z^18 - 5*z^17 + 4*z^15 + 5*z^12 - 5*z^10 + 4*z^6 - 5*z^4 -
1)*x_2^2*x_5,
x_0*x_1*x_3 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 +
2*z^4 - 2)*x_0*x_2*x_3 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 + 3*z^14 -
2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 + 3*z - 2)*x_0*x_2*x_4 +
1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 -
1)*x_0*x_2*x_5 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 +
z^10 - 3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z - 1)*x_1^2*x_4 + 1/3*(z^23 -
2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 - 1)*x_1^2*x_5 +
(z^23 - z^18 + z^17 - z^15 + z^14 - z^12 + z^10 - z^9 - z^6 + z^4 - z^3
+ z - 1)*x_1*x_2*x_3 + x_1*x_2*x_4 + (-z^23 - z^17 - z^14 + z^12 - z^10
+ z^9 - z^4 + z^3 - z + 1)*x_1*x_2*x_5 + 1/3*(-z^23 - z^18 - z^17 - z^15
+ z^12 - z^10 - z^6 - z^4 + 1)*x_2^2*x_3 + 1/3*(2*z^23 - z^18 + 2*z^17 -
z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 - 2)*x_2^2*x_4 + 1/3*(-2*z^23 +
z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1)*x_2^2*x_5,
x_0*x_1*x_4 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 +
2*z^4 + 1)*x_0*x_2*x_3 + 1/3*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 -
z^10 + 2*z^6 - z^4 + 1)*x_0*x_2*x_4 + 1/3*(z^23 + z^18 + z^17 + z^15 -
z^12 + z^10 + z^6 + z^4 + 2)*x_0*x_2*x_5 + x_1^2*x_3 + 1/3*(-2*z^23 +
z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1)*x_1^2*x_4 +
1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 - 1)*x_1^2*x_5
+ (z^18 + z^15 + z^6)*x_1*x_2*x_3 + (-z^18 - z^15 - z^6 - 1)*x_1*x_2*x_5
+ 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 -
2)*x_2^2*x_3 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 -
z^6 + 2*z^4 + 1)*x_2^2*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 -
3*z^14 + 2*z^12 - 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z -
1)*x_2^2*x_5,
x_0*x_1*x_5 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4
- 1)*x_0*x_2*x_3 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 +
z^10 - 3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z - 1)*x_0*x_2*x_4 + 1/3*(2*z^23
- z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 + 1)*x_0*x_2*x_5
+ 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 +
1)*x_1^2*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4
+ 1)*x_1^2*x_5 + (-z^18 - z^15 + z^14 - z^9 - z^6 - z^3 + z)*x_1*x_2*x_3
- x_1*x_2*x_4 + (-z^23 + z^18 - z^17 + z^15 - z^14 + z^12 - z^10 + z^9 +
z^6 - z^4 + z^3 - z + 1)*x_1*x_2*x_5 + 1/3*(-2*z^23 + z^18 - 2*z^17 +
z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 + 2)*x_2^2*x_3 + 1/3*(z^23 - 2*z^18
+ z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 - 1)*x_2^2*x_4 + 1/3*(-z^23
- z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 - 2)*x_2^2*x_5,
x_0*x_3^2 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 -
3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z - 1)*x_0*x_4^2 + 1/3*(2*z^23 - 4*z^18
+ 2*z^17 - 4*z^15 - 2*z^12 + 2*z^10 - 4*z^6 + 2*z^4 - 2)*x_0*x_4*x_5 +
1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 +
2)*x_0*x_5^2 + 1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 -
z^6 + 2*z^4 - 2)*x_2*x_3^2 + 1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15 +
2*z^12 - 2*z^10 - 2*z^6 - 2*z^4 + 2)*x_2*x_3*x_5 + 1/3*(2*z^23 - z^18 +
2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 + 1)*x_2*x_4^2 +
1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15 + 2*z^12 - 2*z^10 - 2*z^6 -
2*z^4 - 4)*x_2*x_4*x_5 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 +
z^6 + z^4 + 2)*x_2*x_5^2,
x_0*x_3*x_4 + 1/6*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 -
2*z^4 - 1)*x_0*x_4^2 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 +
z^6 + z^4 - 1)*x_0*x_4*x_5 + 1/6*(-2*z^23 + z^18 - 2*z^17 + z^15 +
2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1)*x_0*x_5^2 + 1/6*(-z^23 + 2*z^18 -
z^17 + 2*z^15 + z^12 - z^10 + 2*z^6 - z^4 + 1)*x_2*x_3^2 + (-z^14 + z^9
+ z^3 - z)*x_2*x_3*x_4 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 -
z^6 - z^4 - 2)*x_2*x_3*x_5 + 1/6*(2*z^23 - z^18 + 2*z^17 - z^15 + 3*z^14
- 2*z^12 + 2*z^10 - 3*z^9 - z^6 + 2*z^4 - 3*z^3 + 3*z + 4)*x_2*x_4^2 +
1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 -
2)*x_2*x_4*x_5 + 1/6*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 +
z^4 + 2)*x_2*x_5^2,
x_0*x_3*x_5 + 1/6*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 +
1)*x_0*x_4^2 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4
+ 1)*x_0*x_4*x_5 + 1/6*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 +
2*z^6 - z^4 + 1)*x_0*x_5^2 + 1/6*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14
- z^12 + z^10 - 3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z - 1)*x_2*x_3^2 -
x_2*x_3*x_4 + 1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 - 3*z^14 + 2*z^12 -
2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z + 2)*x_2*x_3*x_5 + 1/6*(z^23
+ z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4 + 2)*x_2*x_4^2 +
1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 - z^12 + z^10 - 2*z^6 + z^4 +
2)*x_2*x_4*x_5 + 1/6*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 -
z^6 + 2*z^4 - 2)*x_2*x_5^2,
x_1*x_3^2 + 1/3*(z^23 - 2*z^18 + z^17 - 2*z^15 + 3*z^14 - z^12 + z^10 -
3*z^9 - 2*z^6 + z^4 - 3*z^3 + 3*z - 1)*x_1*x_4^2 + 1/3*(2*z^23 - 4*z^18
+ 2*z^17 - 4*z^15 - 2*z^12 + 2*z^10 - 4*z^6 + 2*z^4 - 2)*x_1*x_4*x_5 +
1/3*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 +
2)*x_1*x_5^2 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 + z^6 + z^4
+ 2)*x_2*x_3^2 + 1/3*(-2*z^23 - 2*z^18 - 2*z^17 - 2*z^15 + 2*z^12 -
2*z^10 - 2*z^6 - 2*z^4 + 2)*x_2*x_3*x_4 + x_2*x_4^2 + 1/3*(-2*z^23 +
4*z^18 - 2*z^17 + 4*z^15 - 6*z^14 + 2*z^12 - 2*z^10 + 6*z^9 + 4*z^6 -
2*z^4 + 6*z^3 - 6*z + 2)*x_2*x_4*x_5 + 1/3*(-z^23 + 2*z^18 - z^17 +
2*z^15 + z^12 - z^10 + 2*z^6 - z^4 + 1)*x_2*x_5^2,
x_1*x_3*x_4 + 1/6*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 -
2*z^4 - 1)*x_1*x_4^2 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 + z^10 +
z^6 + z^4 - 1)*x_1*x_4*x_5 + 1/6*(-2*z^23 + z^18 - 2*z^17 + z^15 +
2*z^12 - 2*z^10 + z^6 - 2*z^4 - 1)*x_1*x_5^2 + 1/6*(-2*z^23 + z^18 -
2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 + 2)*x_2*x_3^2 +
1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 -
2)*x_2*x_3*x_4 - x_2*x_3*x_5 + 1/2*(z^14 - z^9 - z^3 + z)*x_2*x_4^2 +
1/3*(2*z^23 - z^18 + 2*z^17 - z^15 - 2*z^12 + 2*z^10 - z^6 + 2*z^4 +
1)*x_2*x_4*x_5 + 1/6*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 -
z^4 + 1)*x_2*x_5^2,
x_1*x_3*x_5 + 1/6*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 +
1)*x_1*x_4^2 + 1/3*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4
+ 1)*x_1*x_4*x_5 + 1/6*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12 - z^10 +
2*z^6 - z^4 + 1)*x_1*x_5^2 + 1/6*(-z^23 + 2*z^18 - z^17 + 2*z^15 + z^12
- z^10 + 2*z^6 - z^4 + 1)*x_2*x_3^2 + 1/3*(-2*z^23 + z^18 - 2*z^17 +
z^15 - 3*z^14 + 2*z^12 - 2*z^10 + 3*z^9 + z^6 - 2*z^4 + 3*z^3 - 3*z +
2)*x_2*x_3*x_4 + 1/2*x_2*x_4^2 + 1/3*(z^23 + z^18 + z^17 + z^15 - z^12 +
z^10 + z^6 + z^4 - 1)*x_2*x_4*x_5 + 1/6*(z^23 + z^18 + z^17 + z^15 -
z^12 + z^10 + z^6 + z^4 - 1)*x_2*x_5^2,
x_3^3 + 1/3*(-4*z^23 + 3*z^18 - 4*z^17 + 3*z^15 - 4*z^14 + 4*z^12 - 4*z^10 +
4*z^9 + 3*z^6 - 4*z^4 + 4*z^3 - 4*z + 7)*x_3*x_4^2 + 1/3*(-6*z^23 +
4*z^18 - 6*z^17 + 4*z^15 - 2*z^14 + 6*z^12 - 6*z^10 + 2*z^9 + 4*z^6 -
6*z^4 + 2*z^3 - 2*z + 10)*x_3*x_4*x_5 + 1/3*(2*z^23 + 2*z^18 + 2*z^17 +
2*z^15 - 4*z^14 - 2*z^12 + 2*z^10 + 4*z^9 + 2*z^6 + 2*z^4 + 4*z^3 - 4*z
+ 6)*x_3*x_5^2 + 1/3*(-z^23 - z^17 + z^12 - z^10 - z^4 + 1)*x_4^3 +
1/3*(z^23 + 3*z^18 + z^17 + 3*z^15 - 4*z^14 - z^12 + z^10 + 4*z^9 +
3*z^6 + z^4 + 4*z^3 - 4*z - 1)*x_4^2*x_5 + 1/3*(-3*z^23 + 6*z^18 -
3*z^17 + 6*z^15 + z^14 + 3*z^12 - 3*z^10 - z^9 + 6*z^6 - 3*z^4 - z^3 + z
- 3)*x_4*x_5^2 + 1/3*(3*z^23 + 3*z^17 + 2*z^14 - 3*z^12 + 3*z^10 - 2*z^9
+ 3*z^4 - 2*z^3 + 2*z - 3)*x_5^3,
x_3^2*x_4 + 1/9*(-2*z^23 - 5*z^18 - 2*z^17 - 5*z^15 + 3*z^14 + 2*z^12 -
2*z^10 - 3*z^9 - 5*z^6 - 2*z^4 - 3*z^3 + 3*z - 1)*x_3*x_4^2 +
1/9*(-4*z^23 + 2*z^18 - 4*z^17 + 2*z^15 + 4*z^12 - 4*z^10 + 2*z^6 -
4*z^4 - 8)*x_3*x_4*x_5 + 1/9*(4*z^23 - 2*z^18 + 4*z^17 - 2*z^15 + 6*z^14
- 4*z^12 + 4*z^10 - 6*z^9 - 2*z^6 + 4*z^4 - 6*z^3 + 6*z + 5)*x_3*x_5^2 +
1/9*(-z^23 - z^18 - z^17 - z^15 + z^12 - z^10 - z^6 - z^4 - 2)*x_4^3 +
1/9*(4*z^23 + z^18 + 4*z^17 + z^15 - 4*z^12 + 4*z^10 + z^6 + 4*z^4 +
8)*x_4^2*x_5 + 1/9*(-4*z^23 - z^18 - 4*z^17 - z^15 - 6*z^14 + 4*z^12 -
4*z^10 + 6*z^9 - z^6 - 4*z^4 + 6*z^3 - 6*z - 5)*x_4*x_5^2 + 1/3*(-z^23 +
z^18 - z^17 + z^15 + z^12 - z^10 + z^6 - z^4 + 1)*x_5^3,
x_3^2*x_5 + 1/9*(-z^23 + 5*z^18 - z^17 + 5*z^15 - 6*z^14 + z^12 - z^10 +
6*z^9 + 5*z^6 - z^4 + 6*z^3 - 6*z + 4)*x_3*x_4^2 + 1/9*(-14*z^23 +
10*z^18 - 14*z^17 + 10*z^15 - 6*z^14 + 14*z^12 - 14*z^10 + 6*z^9 +
10*z^6 - 14*z^4 + 6*z^3 - 6*z + 2)*x_3*x_4*x_5 + 1/9*(2*z^23 + 8*z^18 +
2*z^17 + 8*z^15 - 2*z^12 + 2*z^10 + 8*z^6 + 2*z^4 + 4)*x_3*x_5^2 +
1/9*(-2*z^23 + z^18 - 2*z^17 + z^15 + 2*z^12 - 2*z^10 + z^6 - 2*z^4 +
5)*x_4^3 + 1/9*(8*z^23 - z^18 + 8*z^17 - z^15 - 3*z^14 - 8*z^12 + 8*z^10
+ 3*z^9 - z^6 + 8*z^4 + 3*z^3 - 3*z - 2)*x_4^2*x_5 + 1/9*(-5*z^23 + z^18
- 5*z^17 + z^15 + 5*z^12 - 5*z^10 + z^6 - 5*z^4 - 13)*x_4*x_5^2 +
1/3*(z^23 - z^18 + z^17 - z^15 + 2*z^14 - z^12 + z^10 - 2*z^9 - z^6 +
z^4 - 2*z^3 + 2*z - 1)*x_5^3
]
The output above shows that this surface is not hyperelliptic. Its canonical ideal is not generated by quadrics, and this surface is not cyclic trigonal. Therefore it is either a plane quintic or a general trigonal surface. We will analyze it further to see if it is a plane quintic; if this were to fail (which it does not) then we could attempt to analyze it as a general trigonal surface.
We use some of the special functions in autcv10e.txt to analyze this potential plane quintic.
The quadrics in the canonical ideal of a plane quintic cut out the Veronese surface in \(\mathbb{P}^5\). Therefore, we first look for all the degree 3 characters \(\psi\) of the group \(G\) such that \( \operatorname{Sym}^2 \psi = \chi_{I_2}\), where \( \chi_{I_2}\) is the character of the \(G\) action on quadrics. This is performed by the PlaneCharacters command.
>T:=CharacterTable(G);
> psis:=PlaneCharacters(T,T[4]+T[7]);
> psis;
[
( 3, 0, 0, -zeta(13)_13^11 - zeta(13)_13^9 - zeta(13)_13^8 - zeta(13)_13^7 -
zeta(13)_13^6 - zeta(13)_13^5 - zeta(13)_13^3 - zeta(13)_13^2 -
zeta(13)_13 - 1, zeta(13)_13^11 + zeta(13)_13^8 + zeta(13)_13^7,
zeta(13)_13^9 + zeta(13)_13^3 + zeta(13)_13, zeta(13)_13^6 +
zeta(13)_13^5 + zeta(13)_13^2 )
]
> psis[1] eq T[5];
true
>
We see that one degree three character satisfies this description. It is the irreducible character $\chi_5$ in the character table above.
Next, for this character \(\psi\), we find the degree 1 characters in \( \operatorname{Sym}^5 \psi\). These are the possible characters of the \(G\) action on the quintic polynomial defining this curve as a subvariety of \(\mathbb{P}^2\).
> Decomposition(T,Symmetrization(psis[1],[5]));
[
1,
1,
1,
2,
1,
2,
1
]
( 0, 0, 0, 0, 0, 0, 0 )
This shows that the quintic must come from the isotypical subspace of \(S_5\) with character \( \chi_1\), \( \chi_2\), or \( \chi_3\).
We begin by computing the \(G\)-invariant quintics.
> G5:=ActionGenerators(GModule(T[5]));
> G5;
[
[1/9*(zeta_13^11 + 2*zeta_13^9 + zeta_13^8 + zeta_13^7 + 6*zeta_13^6 +
6*zeta_13^5 + 2*zeta_13^3 + 6*zeta_13^2 + 2*zeta_13 + 2)
1/9*(-5*zeta_13^11 - zeta_13^9 - 5*zeta_13^8 - 5*zeta_13^7 - 3*zeta_13^6
- 3*zeta_13^5 - zeta_13^3 - 3*zeta_13^2 - zeta_13 - 1)
1/9*(-2*zeta_13^11 - 4*zeta_13^9 - 2*zeta_13^8 - 2*zeta_13^7 -
3*zeta_13^6 - 3*zeta_13^5 - 4*zeta_13^3 - 3*zeta_13^2 - 4*zeta_13 - 4)]
[1/9*(zeta_13^11 + 2*zeta_13^9 + zeta_13^8 + zeta_13^7 - 3*zeta_13^6 -
3*zeta_13^5 + 2*zeta_13^3 - 3*zeta_13^2 + 2*zeta_13 + 2)
1/9*(4*zeta_13^11 - zeta_13^9 + 4*zeta_13^8 + 4*zeta_13^7 - 3*zeta_13^6
- 3*zeta_13^5 - zeta_13^3 - 3*zeta_13^2 - zeta_13 - 1)
1/9*(-2*zeta_13^11 - 4*zeta_13^9 - 2*zeta_13^8 - 2*zeta_13^7 -
3*zeta_13^6 - 3*zeta_13^5 - 4*zeta_13^3 - 3*zeta_13^2 - 4*zeta_13 - 4)]
[1/9*(-2*zeta_13^11 - 4*zeta_13^9 - 2*zeta_13^8 - 2*zeta_13^7 - 3*zeta_13^6
- 3*zeta_13^5 - 4*zeta_13^3 - 3*zeta_13^2 - 4*zeta_13 - 4)
1/9*(zeta_13^11 + 2*zeta_13^9 + zeta_13^8 + zeta_13^7 + 6*zeta_13^6 +
6*zeta_13^5 + 2*zeta_13^3 + 6*zeta_13^2 + 2*zeta_13 + 2)
1/9*(-5*zeta_13^11 - zeta_13^9 - 5*zeta_13^8 - 5*zeta_13^7 - 3*zeta_13^6
- 3*zeta_13^5 - zeta_13^3 - 3*zeta_13^2 - zeta_13 - 1)],
[1/9*(zeta_13^11 + 2*zeta_13^9 + zeta_13^8 + zeta_13^7 - 3*zeta_13^6 -
3*zeta_13^5 + 2*zeta_13^3 - 3*zeta_13^2 + 2*zeta_13 + 2)
1/9*(4*zeta_13^11 - zeta_13^9 + 4*zeta_13^8 + 4*zeta_13^7 - 3*zeta_13^6
- 3*zeta_13^5 - zeta_13^3 - 3*zeta_13^2 - zeta_13 - 1)
1/9*(-2*zeta_13^11 - 4*zeta_13^9 - 2*zeta_13^8 - 2*zeta_13^7 -
3*zeta_13^6 - 3*zeta_13^5 - 4*zeta_13^3 - 3*zeta_13^2 - 4*zeta_13 - 4)]
[1/9*(5*zeta_13^11 + zeta_13^9 + 5*zeta_13^8 + 5*zeta_13^7 + 3*zeta_13^6 +
3*zeta_13^5 + zeta_13^3 + 3*zeta_13^2 + zeta_13 + 1) 1/9*(2*zeta_13^11 +
4*zeta_13^9 + 2*zeta_13^8 + 2*zeta_13^7 + 3*zeta_13^6 + 3*zeta_13^5 +
4*zeta_13^3 + 3*zeta_13^2 + 4*zeta_13 - 5) 1/9*(-zeta_13^11 -
2*zeta_13^9 - zeta_13^8 - zeta_13^7 + 3*zeta_13^6 + 3*zeta_13^5 -
2*zeta_13^3 + 3*zeta_13^2 - 2*zeta_13 - 2)]
[1/3*(-zeta_13^11 + zeta_13^9 - zeta_13^8 - zeta_13^7 + zeta_13^3 + zeta_13
+ 1) 1/3*(-zeta_13^11 + zeta_13^9 - zeta_13^8 - zeta_13^7 + zeta_13^3 +
zeta_13 + 1) 1/3*(-zeta_13^11 + zeta_13^9 - zeta_13^8 - zeta_13^7 +
zeta_13^3 + zeta_13 + 1)]
]
> K<z_39>:=CyclotomicField(39);
> GL3K:=GeneralLinearGroup(3,K);
> rho5:=homGL3K | G5>;
> S<y_0,y_1,y_2>:=PolynomialRing(K,3);
> rho5(G.1 * G.2^5);
[1/9*(4*z_39^23 - 7*z_39^18 + 4*z_39^17 - 7*z_39^15 + 5*z_39^14 - 4*z_39^12 +
4*z_39^10 - 5*z_39^9 - 7*z_39^6 + 4*z_39^4 - 5*z_39^3 + 5*z_39 - 5)
1/9*(-2*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 -
2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)
1/9*(z_39^23 - 4*z_39^18 + z_39^17 - 4*z_39^15 - z_39^14 - z_39^12 + z_39^10
+ z_39^9 - 4*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)]
[1/9*(z_39^23 + 5*z_39^18 + z_39^17 + 5*z_39^15 - z_39^14 - z_39^12 + z_39^10 +
z_39^9 + 5*z_39^6 + z_39^4 + z_39^3 - z_39 + 1) 1/9*(-5*z_39^23 + 2*z_39^18
- 5*z_39^17 + 2*z_39^15 - 4*z_39^14 + 5*z_39^12 - 5*z_39^10 + 4*z_39^9 +
2*z_39^6 - 5*z_39^4 + 4*z_39^3 - 4*z_39 + 4) 1/9*(-2*z_39^23 - z_39^18 -
2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 - 2*z_39^9 - z_39^6
- 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)]
[1/9*(-5*z_39^23 + 2*z_39^18 - 5*z_39^17 + 2*z_39^15 - 4*z_39^14 + 5*z_39^12 -
5*z_39^10 + 4*z_39^9 + 2*z_39^6 - 5*z_39^4 + 4*z_39^3 - 4*z_39 + 4)
1/9*(-2*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 -
2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)
1/9*(z_39^23 + 5*z_39^18 + z_39^17 + 5*z_39^15 - z_39^14 - z_39^12 + z_39^10
+ z_39^9 + 5*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)]
> rho5(G.1^2 * G.2^5);
[1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10 +
2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 + 2) 1/3*(-z_39^23 + z_39^18
- z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10 + 2*z_39^9 + z_39^6 -
z_39^4 + 2*z_39^3 - 2*z_39 + 2) 1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15
- 2*z_39^14 + z_39^12 - z_39^10 + 2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 -
2*z_39 + 2)]
[1/9*(z_39^23 - 4*z_39^18 + z_39^17 - 4*z_39^15 - z_39^14 - z_39^12 + z_39^10 +
z_39^9 - 4*z_39^6 + z_39^4 + z_39^3 - z_39 + 1) 1/9*(4*z_39^23 - 7*z_39^18 +
4*z_39^17 - 7*z_39^15 + 5*z_39^14 - 4*z_39^12 + 4*z_39^10 - 5*z_39^9 -
7*z_39^6 + 4*z_39^4 - 5*z_39^3 + 5*z_39 - 5) 1/9*(-2*z_39^23 - z_39^18 -
2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 - 2*z_39^9 - z_39^6
- 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)]
[1/9*(5*z_39^23 - 2*z_39^18 + 5*z_39^17 - 2*z_39^15 + 4*z_39^14 - 5*z_39^12 +
5*z_39^10 - 4*z_39^9 - 2*z_39^6 + 5*z_39^4 - 4*z_39^3 + 4*z_39 - 4)
1/9*(2*z_39^23 + z_39^18 + 2*z_39^17 + z_39^15 - 2*z_39^14 - 2*z_39^12 +
2*z_39^10 + 2*z_39^9 + z_39^6 + 2*z_39^4 + 2*z_39^3 - 2*z_39 - 7)
1/9*(-z_39^23 + 4*z_39^18 - z_39^17 + 4*z_39^15 + z_39^14 + z_39^12 -
z_39^10 - z_39^9 + 4*z_39^6 - z_39^4 - z_39^3 + z_39 - 1)]
> f5:=IsotypicalSubspace(G,rho5,T,S,5,1);
CharacterRow 1
Dimension 1
Multiplicity 1
> f5;
[
y_0^5 + 1/3*(5*z_39^23 - 5*z_39^18 + 5*z_39^17 - 5*z_39^15 + 5*z_39^14 -
5*z_39^12 + 5*z_39^10 - 5*z_39^9 - 5*z_39^6 + 5*z_39^4 - 5*z_39^3 +
5*z_39 - 5)*y_0^4*y_1 + 1/3*(5*z_39^18 + 5*z_39^15 + 5*z_39^6)*y_0^4*y_2
+ 1/3*(4*z_39^18 + 4*z_39^15 - 6*z_39^14 + 6*z_39^9 + 4*z_39^6 +
6*z_39^3 - 6*z_39)*y_0^3*y_1^2 + 1/3*(-4*z_39^23 - 4*z_39^17 -
12*z_39^14 + 4*z_39^12 - 4*z_39^10 + 12*z_39^9 - 4*z_39^4 + 12*z_39^3 -
12*z_39 + 12)*y_0^3*y_1*y_2 + 1/3*(-2*z_39^23 - 4*z_39^18 - 2*z_39^17 -
4*z_39^15 + 4*z_39^14 + 2*z_39^12 - 2*z_39^10 - 4*z_39^9 - 4*z_39^6 -
2*z_39^4 - 4*z_39^3 + 4*z_39 - 4)*y_0^3*y_2^2 + 1/3*(-2*z_39^23 -
4*z_39^18 - 2*z_39^17 - 4*z_39^15 + 4*z_39^14 + 2*z_39^12 - 2*z_39^10 -
4*z_39^9 - 4*z_39^6 - 2*z_39^4 - 4*z_39^3 + 4*z_39 - 4)*y_0^2*y_1^3 +
(4*z_39^23 - 2*z_39^18 + 4*z_39^17 - 2*z_39^15 - 4*z_39^12 + 4*z_39^10 -
2*z_39^6 + 4*z_39^4 - 4)*y_0^2*y_1^2*y_2 + (4*z_39^23 - 2*z_39^18 +
4*z_39^17 - 2*z_39^15 - 4*z_39^12 + 4*z_39^10 - 2*z_39^6 + 4*z_39^4 -
4)*y_0^2*y_1*y_2^2 + 1/3*(4*z_39^18 + 4*z_39^15 - 6*z_39^14 + 6*z_39^9 +
4*z_39^6 + 6*z_39^3 - 6*z_39)*y_0^2*y_2^3 + 1/3*(5*z_39^18 + 5*z_39^15 +
5*z_39^6)*y_0*y_1^4 + 1/3*(-4*z_39^23 - 4*z_39^17 - 12*z_39^14 +
4*z_39^12 - 4*z_39^10 + 12*z_39^9 - 4*z_39^4 + 12*z_39^3 - 12*z_39 +
12)*y_0*y_1^3*y_2 + (4*z_39^23 - 2*z_39^18 + 4*z_39^17 - 2*z_39^15 -
4*z_39^12 + 4*z_39^10 - 2*z_39^6 + 4*z_39^4 - 4)*y_0*y_1^2*y_2^2 +
1/3*(-4*z_39^23 - 4*z_39^17 - 12*z_39^14 + 4*z_39^12 - 4*z_39^10 +
12*z_39^9 - 4*z_39^4 + 12*z_39^3 - 12*z_39 + 12)*y_0*y_1*y_2^3 +
1/3*(5*z_39^23 - 5*z_39^18 + 5*z_39^17 - 5*z_39^15 + 5*z_39^14 -
5*z_39^12 + 5*z_39^10 - 5*z_39^9 - 5*z_39^6 + 5*z_39^4 - 5*z_39^3 +
5*z_39 - 5)*y_0*y_2^4 + y_1^5 + 1/3*(5*z_39^23 - 5*z_39^18 + 5*z_39^17 -
5*z_39^15 + 5*z_39^14 - 5*z_39^12 + 5*z_39^10 - 5*z_39^9 - 5*z_39^6 +
5*z_39^4 - 5*z_39^3 + 5*z_39 - 5)*y_1^4*y_2 + 1/3*(4*z_39^18 + 4*z_39^15
- 6*z_39^14 + 6*z_39^9 + 4*z_39^6 + 6*z_39^3 - 6*z_39)*y_1^3*y_2^2 +
1/3*(-2*z_39^23 - 4*z_39^18 - 2*z_39^17 - 4*z_39^15 + 4*z_39^14 +
2*z_39^12 - 2*z_39^10 - 4*z_39^9 - 4*z_39^6 - 2*z_39^4 - 4*z_39^3 +
4*z_39 - 4)*y_1^2*y_2^3 + 1/3*(5*z_39^18 + 5*z_39^15 +
5*z_39^6)*y_1*y_2^4 + y_2^5
]
In the next section we check that this polynomial defines a smooth plane quintic with the correct automorphisms.
> K<z_39>:=CyclotomicField(39);
> P2<y_0,y_1,y_2>:=ProjectiveSpace(K,2);
> X:=Scheme(P2,[
> y_0^5 + 1/3*(5*z_39^23 - 5*z_39^18 + 5*z_39^17 - 5*z_39^15 + 5*z_39^14 - 5*z\
_39^12 + 5*z_39^10 - 5*z_39^9 - 5*z_39^6 + 5*z_39^4 - 5*z_39^3 + 5*z_39 -5)*y_\
0^4*y_1 + 1/3*(5*z_39^18 + 5*z_39^15 + 5*z_39^6)*y_0^4*y_2 + 1/3*(4*z_39^18 + \
4*z_39^15 - 6*z_39^14 + 6*z_39^9 + 4*z_39^6 + 6*z_39^3 - 6*z_39)*y_0^3*y_1^2 +\
1/3*(-4*z_39^23 - 4*z_39^17 - 12*z_39^14 + 4*z_39^12 -4*z_39^10 + 12*z_39^9 -\
4*z_39^4 + 12*z_39^3 - 12*z_39 + 12)*y_0^3*y_1*y_2 +1/3*(-2*z_39^23 - 4*z_39^\
18 - 2*z_39^17 - 4*z_39^15 + 4*z_39^14 + 2*z_39^12 - 2*z_39^10 - 4*z_39^9 - 4*\
z_39^6 - 2*z_39^4 - 4*z_39^3 + 4*z_39 - 4)*y_0^3*y_2^2 + 1/3*(-2*z_39^23 - 4*z\
_39^18 - 2*z_39^17 - 4*z_39^15 + 4*z_39^14 + 2*z_39^12 - 2*z_39^10 - 4*z_39^9 \
- 4*z_39^6 - 2*z_39^4 - 4*z_39^3 + 4*z_39 - 4)*y_0^2*y_1^3 + (4*z_39^23 - 2*z_\
39^18 + 4*z_39^17 - 2*z_39^15 - 4*z_39^12 + 4*z_39^10 - 2*z_39^6 + 4*z_39^4 - \
4)*y_0^2*y_1^2*y_2+ (4*z_39^23 - 2*z_39^18 + 4*z_39^17 - 2*z_39^15 - 4*z_39^12\
+ 4*z_39^10 - 2*z_39^6 + 4*z_39^4 - 4)*y_0^2*y_1*y_2^2 + 1/3*(4*z_39^18 + 4*z\
_39^15 - 6*z_39^14 + 6*z_39^9 + 4*z_39^6 + 6*z_39^3 - 6*z_39)*y_0^2*y_2^3 + 1/\
3*(5*z_39^18 + 5*z_39^15 + 5*z_39^6)*y_0*y_1^4 + 1/3*(-4*z_39^23 - 4*z_39^17 -\
12*z_39^14 + 4*z_39^12 - 4*z_39^10 + 12*z_39^9 - 4*z_39^4 + 12*z_39^3 - 12*z_\
39 + 12)*y_0*y_1^3*y_2 + (4*z_39^23 - 2*z_39^18 + 4*z_39^17- 2*z_39^15 - 4*z_3\
9^12 + 4*z_39^10 - 2*z_39^6 + 4*z_39^4 - 4)*y_0*y_1^2*y_2^2 + 1/3*(-4*z_39^23 \
- 4*z_39^17 - 12*z_39^14 + 4*z_39^12 - 4*z_39^10 + 12*z_39^9 - 4*z_39^4 + 12*z\
_39^3 - 12*z_39 + 12)*y_0*y_1*y_2^3 +1/3*(5*z_39^23 - 5*z_39^18 + 5*z_39^17 - \
5*z_39^15 + 5*z_39^14 - 5*z_39^12 +5*z_39^10 - 5*z_39^9 - 5*z_39^6 + 5*z_39^4 \
- 5*z_39^3 + 5*z_39 - 5)*y_0*y_2^4 + y_1^5 + 1/3*(5*z_39^23 - 5*z_39^18 + 5*z_\
39^17 - 5*z_39^15 + 5*z_39^14 - 5*z_39^12 + 5*z_39^10 - 5*z_39^9 - 5*z_39^6 + \
5*z_39^4 - 5*z_39^3 + 5*z_39 - 5)*y_1^4*y_2 + 1/3*(4*z_39^18 + 4*z_39^15 - 6*z\
_39^14 + 6*z_39^9 + 4*z_39^6 + 6*z_39^3 - 6*z_39)*y_1^3*y_2^2 + 1/3*(-2*z_39^2\
3 - 4*z_39^18 - 2*z_39^17 - 4*z_39^15 + 4*z_39^14 + 2*z_39^12 - 2*z_39^10 - 4*\
z_39^9 - 4*z_39^6 - 2*z_39^4 - 4*z_39^3 + 4*z_39 - 4)*y_1^2*y_2^3 + 1/3*(5*z_3\
9^18 + 5*z_39^15 + 5*z_39^6)*y_1*y_2^4 + y_2^5]);
> Dimension(X);
1
> IsSingular(X);
false
> A:=Matrix([
> [1/9*(4*z_39^23 - 7*z_39^18 + 4*z_39^17 - 7*z_39^15 + 5*z_39^14 - 4*z_39^12 \
+ 4*z_39^10 - 5*z_39^9 - 7*z_39^6 + 4*z_39^4 - 5*z_39^3 + 5*z_39 - 5), 1/9*(-2\
*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 -\
2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2),1/9*(z_39^23 - 4*z_39^1\
8 + z_39^17 - 4*z_39^15 - z_39^14 - z_39^12 + z_39^10+ z_39^9 - 4*z_39^6 + z_3\
9^4 + z_39^3 - z_39 + 1)],
> [1/9*(z_39^23 + 5*z_39^18 + z_39^17 + 5*z_39^15 - z_39^14 - z_39^12 + z_39^1\
0 + z_39^9 + 5*z_39^6 + z_39^4 + z_39^3 - z_39 + 1), 1/9*(-5*z_39^23 + 2*z_39^\
18 - 5*z_39^17 + 2*z_39^15 - 4*z_39^14 + 5*z_39^12 - 5*z_39^10 + 4*z_39^9 + 2*\
z_39^6 - 5*z_39^4 + 4*z_39^3 - 4*z_39 + 4), 1/9*(-2*z_39^23 - z_39^18 - 2*z_39\
^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39\
^4 - 2*z_39^3 + 2*z_39 - 2)],
> [1/9*(-5*z_39^23 + 2*z_39^18 - 5*z_39^17 + 2*z_39^15 - 4*z_39^14 + 5*z_39^12\
- 5*z_39^10 + 4*z_39^9 + 2*z_39^6 - 5*z_39^4 + 4*z_39^3 - 4*z_39 + 4), 1/9*(-\
2*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 \
- 2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2), 1/9*(z_39^23 + 5*z_39\
^18 + z_39^17 + 5*z_39^15 - z_39^14 - z_39^12 + z_39^10+ z_39^9 + 5*z_39^6 + z\
_39^4 + z_39^3 - z_39 + 1)]
> ]);
> B:=Matrix([
> [1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10\
+ 2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 + 2), 1/3*(-z_39^23 + z_39^1\
8 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10 + 2*z_39^9 + z_39^6 - z_\
39^4 + 2*z_39^3 - 2*z_39 + 2), 1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15 - 2\
*z_39^14 + z_39^12 - z_39^10 + 2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 \
+ 2)],
> [1/9*(z_39^23 - 4*z_39^18 + z_39^17 - 4*z_39^15 - z_39^14 - z_39^12 + z_39^1\
0 + z_39^9 - 4*z_39^6 + z_39^4 + z_39^3 - z_39 + 1), 1/9*(4*z_39^23 - 7*z_39^1\
8 +4*z_39^17 - 7*z_39^15 + 5*z_39^14 - 4*z_39^12 + 4*z_39^10 - 5*z_39^9 - 7*z_\
39^6 + 4*z_39^4 - 5*z_39^3 + 5*z_39 - 5), 1/9*(-2*z_39^23 - z_39^18 - 2*z_39^1\
7 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39^4\
- 2*z_39^3 + 2*z_39 - 2)],
> [1/9*(5*z_39^23 - 2*z_39^18 + 5*z_39^17 - 2*z_39^15 + 4*z_39^14 - 5*z_39^12 \
+ 5*z_39^10 - 4*z_39^9 - 2*z_39^6 + 5*z_39^4 - 4*z_39^3 + 4*z_39 - 4), 1/9*(2*\
z_39^23 + z_39^18 + 2*z_39^17 + z_39^15 - 2*z_39^14 - 2*z_39^12 + 2*z_39^10 + \
2*z_39^9 + z_39^6 + 2*z_39^4 + 2*z_39^3 - 2*z_39 - 7), 1/9*(-z_39^23 + 4*z_39^\
18 - z_39^17 + 4*z_39^15 + z_39^14 + z_39^12 - z_39^10 - z_39^9 + 4*z_39^6 - z\
_39^4 - z_39^3 + z_39 - 1)]
> ]);
> GL3K:=GeneralLinearGroup(3,K);
> Order(A);
3
> Order(B);
3
> Order( (A*B)^(-1));
13
> IdentifyGroup(sub<GL3K | A,B>);
<39, 1>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
1/9*(4*z_39^23 - 7*z_39^18 + 4*z_39^17 - 7*z_39^15 + 5*z_39^14 - 4*z_39^12 +
4*z_39^10 - 5*z_39^9 - 7*z_39^6 + 4*z_39^4 - 5*z_39^3 + 5*z_39 - 5)*y_0 +
1/9*(z_39^23 + 5*z_39^18 + z_39^17 + 5*z_39^15 - z_39^14 - z_39^12 + z_39^10
+ z_39^9 + 5*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)*y_1 + 1/9*(-5*z_39^23 +
2*z_39^18 - 5*z_39^17 + 2*z_39^15 - 4*z_39^14 + 5*z_39^12 - 5*z_39^10 +
4*z_39^9 + 2*z_39^6 - 5*z_39^4 + 4*z_39^3 - 4*z_39 + 4)*y_2
1/9*(-2*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 -
2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)*y_0 +
1/9*(-5*z_39^23 + 2*z_39^18 - 5*z_39^17 + 2*z_39^15 - 4*z_39^14 + 5*z_39^12
- 5*z_39^10 + 4*z_39^9 + 2*z_39^6 - 5*z_39^4 + 4*z_39^3 - 4*z_39 + 4)*y_1 +
1/9*(-2*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 -
2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)*y_2
1/9*(z_39^23 - 4*z_39^18 + z_39^17 - 4*z_39^15 - z_39^14 - z_39^12 + z_39^10 +
z_39^9 - 4*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)*y_0 + 1/9*(-2*z_39^23 -
z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 - 2*z_39^9
- z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)*y_1 + 1/9*(z_39^23 + 5*z_39^18
+ z_39^17 + 5*z_39^15 - z_39^14 - z_39^12 + z_39^10 + z_39^9 + 5*z_39^6 +
z_39^4 + z_39^3 - z_39 + 1)*y_2
and inverse
1/9*(-z_39^23 + 4*z_39^18 - z_39^17 + 4*z_39^15 + z_39^14 + z_39^12 - z_39^10 -
z_39^9 + 4*z_39^6 - z_39^4 - z_39^3 + z_39 - 1)*y_0 + 1/3*(-z_39^23 +
z_39^18 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10 + 2*z_39^9 +
z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 + 2)*y_1 + 1/9*(-2*z_39^23 - z_39^18 -
2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 - 2*z_39^9 - z_39^6
- 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)*y_2
1/9*(5*z_39^23 - 2*z_39^18 + 5*z_39^17 - 2*z_39^15 + 4*z_39^14 - 5*z_39^12 +
5*z_39^10 - 4*z_39^9 - 2*z_39^6 + 5*z_39^4 - 4*z_39^3 + 4*z_39 - 4)*y_0 +
1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10
+ 2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 + 2)*y_1 + 1/9*(z_39^23 -
4*z_39^18 + z_39^17 - 4*z_39^15 - z_39^14 - z_39^12 + z_39^10 + z_39^9 -
4*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)*y_2
1/9*(2*z_39^23 + z_39^18 + 2*z_39^17 + z_39^15 - 2*z_39^14 - 2*z_39^12 +
2*z_39^10 + 2*z_39^9 + z_39^6 + 2*z_39^4 + 2*z_39^3 - 2*z_39 - 7)*y_0 +
1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10
+ 2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 + 2)*y_1 + 1/9*(4*z_39^23 -
7*z_39^18 + 4*z_39^17 - 7*z_39^15 + 5*z_39^14 - 4*z_39^12 + 4*z_39^10 -
5*z_39^9 - 7*z_39^6 + 4*z_39^4 - 5*z_39^3 + 5*z_39 - 5)*y_2
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10 +
2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 + 2)*y_0 + 1/9*(z_39^23 -
4*z_39^18 + z_39^17 - 4*z_39^15 - z_39^14 - z_39^12 + z_39^10 + z_39^9 -
4*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)*y_1 + 1/9*(5*z_39^23 - 2*z_39^18 +
5*z_39^17 - 2*z_39^15 + 4*z_39^14 - 5*z_39^12 + 5*z_39^10 - 4*z_39^9 -
2*z_39^6 + 5*z_39^4 - 4*z_39^3 + 4*z_39 - 4)*y_2
1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10 +
2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 + 2)*y_0 + 1/9*(4*z_39^23 -
7*z_39^18 + 4*z_39^17 - 7*z_39^15 + 5*z_39^14 - 4*z_39^12 + 4*z_39^10 -
5*z_39^9 - 7*z_39^6 + 4*z_39^4 - 5*z_39^3 + 5*z_39 - 5)*y_1 + 1/9*(2*z_39^23
+ z_39^18 + 2*z_39^17 + z_39^15 - 2*z_39^14 - 2*z_39^12 + 2*z_39^10 +
2*z_39^9 + z_39^6 + 2*z_39^4 + 2*z_39^3 - 2*z_39 - 7)*y_2
1/3*(-z_39^23 + z_39^18 - z_39^17 + z_39^15 - 2*z_39^14 + z_39^12 - z_39^10 +
2*z_39^9 + z_39^6 - z_39^4 + 2*z_39^3 - 2*z_39 + 2)*y_0 + 1/9*(-2*z_39^23 -
z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 - 2*z_39^10 - 2*z_39^9
- z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)*y_1 + 1/9*(-z_39^23 + 4*z_39^18
- z_39^17 + 4*z_39^15 + z_39^14 + z_39^12 - z_39^10 - z_39^9 + 4*z_39^6 -
z_39^4 - z_39^3 + z_39 - 1)*y_2
and inverse
1/9*(-5*z_39^23 + 2*z_39^18 - 5*z_39^17 + 2*z_39^15 - 4*z_39^14 + 5*z_39^12 -
5*z_39^10 + 4*z_39^9 + 2*z_39^6 - 5*z_39^4 + 4*z_39^3 - 4*z_39 + 4)*y_0 +
1/9*(-2*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 -
2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)*y_1 +
1/9*(-2*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 -
2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)*y_2
1/9*(-2*z_39^23 - z_39^18 - 2*z_39^17 - z_39^15 + 2*z_39^14 + 2*z_39^12 -
2*z_39^10 - 2*z_39^9 - z_39^6 - 2*z_39^4 - 2*z_39^3 + 2*z_39 - 2)*y_0 +
1/9*(z_39^23 + 5*z_39^18 + z_39^17 + 5*z_39^15 - z_39^14 - z_39^12 + z_39^10
+ z_39^9 + 5*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)*y_1 + 1/9*(z_39^23 -
4*z_39^18 + z_39^17 - 4*z_39^15 - z_39^14 - z_39^12 + z_39^10 + z_39^9 -
4*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)*y_2
1/9*(z_39^23 + 5*z_39^18 + z_39^17 + 5*z_39^15 - z_39^14 - z_39^12 + z_39^10 +
z_39^9 + 5*z_39^6 + z_39^4 + z_39^3 - z_39 + 1)*y_0 + 1/9*(-5*z_39^23 +
2*z_39^18 - 5*z_39^17 + 2*z_39^15 - 4*z_39^14 + 5*z_39^12 - 5*z_39^10 +
4*z_39^9 + 2*z_39^6 - 5*z_39^4 + 4*z_39^3 - 4*z_39 + 4)*y_1 + 1/9*(4*z_39^23
- 7*z_39^18 + 4*z_39^17 - 7*z_39^15 + 5*z_39^14 - 4*z_39^12 + 4*z_39^10 -
5*z_39^9 - 7*z_39^6 + 4*z_39^4 - 5*z_39^3 + 5*z_39 - 5)*y_2
> K<z_13>:=CyclotomicField(13);
> P2<y_0,y_1,y_2>:=ProjectiveSpace(K,2);
> X:=Scheme(P2,[y_0^4*y_1+y_1^4*y_2+y_2^4*y_0]);
> IsSingular(X);
false
> A:=Matrix([
> [0,0,z_13^12],
> [z_13^4,0,0],
> [0,z_13^10,0]
> ]);
> B:=Matrix([
> [0,z_13^7,0],
> [0,0,z_13^11],
> [z_13^8,0,0]
> ]);
> Order(A);
3
> Order(B);
3
> Order( (A*B)^(-1));
13
> GL3K:=GeneralLinearGroup(3,K);
> IdentifyGroup(sub<GL3K | A,B>);
<39, 1>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
z_13^4*y_1
z_13^10*y_2
(-z_13^11 - z_13^10 - z_13^9 - z_13^8 - z_13^7 - z_13^6 - z_13^5 - z_13^4 -
z_13^3 - z_13^2 - z_13 - 1)*y_0
and inverse
z_13*y_2
z_13^9*y_0
z_13^3*y_1
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
z_13^8*y_2
z_13^7*y_0
z_13^11*y_1
and inverse
z_13^6*y_1
z_13^2*y_2
z_13^5*y_0