|
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 Thomas Breuer.
They list a genus 6 Riemann surface with automorphism group (150,5) in the GAP library of small groups. (We will see that it is the Fermat quintic.) The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (2,3,10).
We use Magma to compute equations of this Riemann surface. 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 Mon Mar 21 2016 16:53:46 on Davids-MacBook-Pro-2 [Seed =
1323845449]
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| Simons Foundation |
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Type ? for help. Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(150,5);
> RunExample(G,6,[2,3,10]);
Set seed to 0.
Character Table of Group G
--------------------------
-------------------------------------------------------------------
Class | 1 2 3 4 5 6 7 8 9 10 11 12 13
Size | 1 15 50 3 3 3 3 6 6 15 15 15 15
Order | 1 2 3 5 5 5 5 5 5 10 10 10 10
-------------------------------------------------------------------
p = 2 1 1 3 6 4 7 5 9 8 5 4 7 6
p = 3 1 2 1 5 7 4 6 9 8 12 10 13 11
p = 5 1 2 3 1 1 1 1 1 1 2 2 2 2
-------------------------------------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1 1 1 1 1
X.2 + 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1
X.3 + 2 0 -1 2 2 2 2 2 2 0 0 0 0
X.4 0 3 1 0 Z1 Z1#3 Z1#2 Z1#4 Z3 Z3#2 Z5 Z5#2 Z5#3 Z5#4
X.5 0 3 1 0 Z1#4 Z1#2 Z1#3 Z1 Z3 Z3#2 Z5#4 Z5#3 Z5#2 Z5
X.6 0 3 -1 0 Z1 Z1#3 Z1#2 Z1#4 Z3 Z3#2 -Z5-Z5#2-Z5#3-Z5#4
X.7 0 3 1 0 Z1#3 Z1#4 Z1 Z1#2 Z3#2 Z3 Z5#3 Z5 Z5#4 Z5#2
X.8 0 3 1 0 Z1#2 Z1 Z1#4 Z1#3 Z3#2 Z3 Z5#2 Z5#4 Z5 Z5#3
X.9 0 3 -1 0 Z1#2 Z1 Z1#4 Z1#3 Z3#2 Z3-Z5#2-Z5#4 -Z5-Z5#3
X.10 0 3 -1 0 Z1#3 Z1#4 Z1 Z1#2 Z3#2 Z3-Z5#3 -Z5-Z5#4-Z5#2
X.11 0 3 -1 0 Z1#4 Z1#2 Z1#3 Z1 Z3 Z3#2-Z5#4-Z5#3-Z5#2 -Z5
X.12 + 6 0 0 Z2 Z2#2 Z2#2 Z2 Z4 Z4#2 0 0 0 0
X.13 + 6 0 0 Z2#2 Z2 Z2 Z2#2 Z4#2 Z4 0 0 0 0
Explanation of Character Value Symbols
--------------------------------------
# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k
Z1 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | 0, 1, 2, 0 ]
Z2 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | 0, 0, -2, -2
]
Z3 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | 1, 0, 1, 1 ]
Z4 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | -1, 0, 1, 1
]
Z5 = (CyclotomicField(5: Sparse := true)) ! [ RationalField() | -1, -1, -1,
-1 ]
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 15
Rep G.1
[3] Order 3 Length 50
Rep G.2
[4] Order 5 Length 3
Rep G.3^3
[5] Order 5 Length 3
Rep G.3^4
[6] Order 5 Length 3
Rep G.3
[7] Order 5 Length 3
Rep G.3^2
[8] Order 5 Length 6
Rep G.3^3 * G.4
[9] Order 5 Length 6
Rep G.4
[10] Order 10 Length 15
Rep G.1 * G.3^4 * G.4^4
[11] Order 10 Length 15
Rep G.1 * G.3^3 * G.4^3
[12] Order 10 Length 15
Rep G.1 * G.3^2 * G.4^2
[13] Order 10 Length 15
Rep G.1 * G.3 * G.4
Surface kernel generators: [ G.1 * G.3 * G.4^2, G.2^2 * G.3^4 * G.4^4, G.1 *
G.2^2 * G.4 ]
Is hyperelliptic? false
Is cyclic trigonal? false
Multiplicities of irreducibles in relevant G-modules:
I_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1 =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0 ]
I_2 =[ 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0 ]
S_2 =[ 0, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0 ]
H^0(C,2K)=[ 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0 ]
I_3 =[ 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 2 ]
S_3 =[ 0, 3, 1, 0, 0, 2, 0, 1, 2, 0, 2, 2, 3 ]
H^0(C,3K)=[ 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1 ]
I2timesS1=[ 0, 2, 2, 0, 0, 1, 0, 0, 0, 0, 1, 1, 3 ]
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 150 and degree 40
[
[z^35 - z^10 -z^35 + z^15 + z^10 - 1 z^30 0 0 0]
[-z^35 + z^15 + z^10 - 1 -z^15 1 0 0 0]
[1 -z^35 - z^30 + z^15 + z^10 - 1 -z^35 + z^15 + z^10 - 1 0 0 0]
[0 0 0 0 0 -z^15]
[0 0 0 z^30 - z^15 -1 -z^35 + z^15 + z^10 - 1]
[0 0 0 z^35 - z^10 0 0],
[0 0 -z^15 0 0 0]
[1 0 0 0 0 0]
[0 z^35 - z^10 0 0 0 0]
[0 0 0 -z^35 - z^30 + z^15 + z^10 - 1 -z^35 - z^30 + z^15 + z^10 - 1 z^15]
[0 0 0 z^35 - z^10 z^35 + z^30 - z^15 - z^10 + 1 z^35 - z^15 - z^10 + 1]
[0 0 0 0 z^35 - z^10 0],
[z^15 - 1 z^30 - z^15 -z^15 + 1 0 0 0]
[-z^35 + z^10 - 1 z^35 + z^30 - z^15 - z^10 + 1 z^35 - z^15 - z^10 + 1 0 0
0]
[0 0 z^35 - z^10 0 0 0]
[0 0 0 z^30 0 0]
[0 0 0 -z^30 -z^15 1]
[0 0 0 0 0 z^30],
[z^35 - z^15 - z^10 + 1 -z^30 + z^15 - 1 -z^35 + z^15 + z^10 - 1 0 0 0]
[z^15 0 -z^35 + z^10 0 0 0]
[z^35 + z^30 - z^15 - z^10 + 1 z^15 0 0 0 0]
[0 0 0 -z^30 + z^15 0 z^35 - z^15 - z^10 + 1]
[0 0 0 z^30 - z^15 z^30 -z^35 + z^10 - 1]
[0 0 0 -z^15 + 1 0 -z^35 + z^10]
]
Matrix Surface Kernel Generators:
[
[-z^30 + z^15 - 1 -z^35 + z^10 - 1 z^30 - z^15 0 0 0]
[z^15 - 1 z^30 - z^15 -z^15 + 1 0 0 0]
[-1 z^35 + z^30 - z^15 - z^10 + 1 0 0 0 0]
[0 0 0 -1 0 -z^30 + z^15 - 1]
[0 0 0 0 -1 -z^35 + z^10]
[0 0 0 0 0 1],
[-z^30 + z^15 -1 -z^35 + z^10 - 1 0 0 0]
[z^35 - z^10 z^30 z^30 - z^15 0 0 0]
[-z^30 + z^15 - 1 z^30 -z^15 0 0 0]
[0 0 0 0 -z^30 z^15]
[0 0 0 -z^30 + z^15 0 z^35 - z^15 - z^10 + 1]
[0 0 0 -z^35 + z^10 -z^35 + z^10 0],
[0 0 z^35 - z^10 0 0 0]
[0 z^30 0 0 0 0]
[-z^35 + z^10 z^35 - z^15 - z^10 + 1 -z^30 + z^15 0 0 0]
[0 0 0 z^30 - z^15 -1 -z^35 + z^15 + z^10 - 1]
[0 0 0 -z^30 + z^15 1 z^35 - z^10 + 1]
[0 0 0 z^15 - 1 z^15 z^35 - z^10]
]
Finding quadrics:
I2 contains a 3-dimensional subspace of CharacterRow 4
Dimension 6
Multiplicity 2
[
x_0^2 - 2*z^30*x_0*x_2 + (z^35 - z^10)*x_2^2,
x_0*x_1 - x_0*x_2 + (z^35 + z^30 - z^15 - z^10 + 1)*x_1*x_2 + 1/2*(z^35 +
z^30 - z^10)*x_2^2,
x_1^2 + 2*z^15*x_1*x_2 + z^30*x_2^2,
x_3^2 + (z^30 - z^15)*x_3*x_5 + (-z^35 - z^30 + z^15 + z^10 - 1)*x_5^2,
x_3*x_4 + z^30*x_4*x_5,
x_4^2 + (z^30 + z^15)*x_4*x_5
]
I2 contains a 3-dimensional subspace of CharacterRow 8
Dimension 6
Multiplicity 2
[
x_0^2 + (z^35 - z^15 - z^10 + 1)*x_0*x_2 + x_2^2,
x_0*x_1 + z^15*x_0*x_2 - z^30*x_1*x_2 + (-z^35 - z^30 + z^15 + z^10 -
1)*x_2^2,
x_0*x_3 + z^30*x_0*x_5 - z^30*x_2*x_3 + (-z^35 + z^10)*x_2*x_5,
x_0*x_4 - z^30*x_1*x_4,
x_1^2 + (z^15 - 1)*x_1*x_2 - z^15*x_2^2,
x_1*x_3 - x_1*x_4 - z^15*x_1*x_5 + z^15*x_2*x_3 - z^15*x_2*x_4 -
z^30*x_2*x_5
]
Finding cubics:
I3 contains a 2-dimensional subspace of CharacterRow 2
Dimension 3
Multiplicity 3
[
x_0^2*x_1 + z^15*x_0^2*x_2 - z^30*x_0*x_1^2 + (-z^35 - 2*z^30 + z^15 + z^10
- 1)*x_0*x_1*x_2 + (-z^35 - z^30 + z^15 + z^10 - 1)*x_0*x_2^2 + (z^35 -
z^10)*x_1^2*x_2 - x_1*x_2^2,
x_0^2*x_3 + z^15*x_0^2*x_4 + z^30*x_0^2*x_5 + (-2*z^35 - 2*z^30 + 2*z^15 +
2*z^10 - 2)*x_0*x_1*x_4 - 2*z^30*x_0*x_2*x_3 + (-2*z^35 +
2*z^10)*x_0*x_2*x_5 + (-z^35 + z^10)*x_1^2*x_3 + (z^35 - z^10 -
1)*x_1^2*x_4 - x_1^2*x_5 + 2*x_1*x_2*x_3 - 2*x_1*x_2*x_4 -
2*z^15*x_1*x_2*x_5 + (z^35 + z^15 - z^10)*x_2^2*x_3 - z^15*x_2^2*x_4 +
(-z^30 - z^15)*x_2^2*x_5,
x_3^2*x_4 - x_3*x_4^2 + (z^30 - z^15)*x_3*x_4*x_5 - z^30*x_4^2*x_5 + (-z^35
- z^30 + z^15 + z^10 - 1)*x_4*x_5^2
]
I3 contains a 2-dimensional subspace of CharacterRow 3
Dimension 2
Multiplicity 1
[
x_0^2*x_3 + z^30*x_0^2*x_5 - 2*z^30*x_0*x_2*x_3 + (-2*z^35 +
2*z^10)*x_0*x_2*x_5 + (z^35 - z^10)*x_1^2*x_3 + (-z^35 + z^10)*x_1^2*x_4
+ x_1^2*x_5 - 2*x_1*x_2*x_3 + 2*x_1*x_2*x_4 + 2*z^15*x_1*x_2*x_5 + (z^35
- z^15 - z^10)*x_2^2*x_3 + z^15*x_2^2*x_4 + (z^30 - z^15)*x_2^2*x_5,
x_0^2*x_4 - 2*z^30*x_0*x_1*x_4 + (z^35 + z^30 - z^15 - z^10 + 1)*x_1^2*x_3 +
(-z^30 + z^15 - 1)*x_1^2*x_4 + (-z^35 + z^10)*x_1^2*x_5 + (2*z^35 -
2*z^10)*x_1*x_2*x_3 + (-2*z^35 + 2*z^10)*x_1*x_2*x_4 + 2*x_1*x_2*x_5 -
x_2^2*x_3 + x_2^2*x_4 + z^15*x_2^2*x_5
]
I3 contains a 3-dimensional subspace of CharacterRow 6
Dimension 6
Multiplicity 2
[
x_0^2*x_3 - x_0^2*x_4 - z^15*x_0^2*x_5 + (2*z^35 + z^30 - z^15 - 2*z^10 +
1)*x_0*x_1*x_5 + (z^35 - z^15 - z^10 + 1)*x_0*x_2*x_3 + (-z^35 + z^15 +
z^10 - 1)*x_0*x_2*x_4 + (z^35 + z^30 - z^15 - z^10)*x_0*x_2*x_5 + (-z^35
+ z^10)*x_1^2*x_3 + z^15*x_1^2*x_5 + (z^35 - z^10 + 1)*x_1*x_2*x_3 +
(z^30 + 1)*x_1*x_2*x_5 - x_2^2*x_4,
x_0*x_1*x_3 + z^30*x_0*x_1*x_5 + z^15*x_0*x_2*x_3 + (z^35 + z^30 - z^15 -
z^10 + 1)*x_0*x_2*x_5 - z^30*x_1^2*x_3 + (-z^35 + z^10)*x_1^2*x_5 +
(-z^35 - z^30 + z^15 + z^10 - 1)*x_1*x_2*x_3 + x_1*x_2*x_5,
x_0*x_1*x_4 + z^15*x_0*x_2*x_4 - z^30*x_1*x_2*x_4 + (-z^35 - z^30 + z^15 +
z^10 - 1)*x_2^2*x_4,
x_0*x_3^2 + 2*z^30*x_0*x_3*x_5 + (z^35 - z^10)*x_0*x_5^2 - z^30*x_2*x_3^2 +
(-2*z^35 + 2*z^10)*x_2*x_3*x_5 + z^15*x_2*x_5^2,
x_0*x_4^2 - z^30*x_1*x_4^2,
x_1*x_3^2 - 2*x_1*x_3*x_4 - 2*z^15*x_1*x_3*x_5 + x_1*x_4^2 +
2*z^15*x_1*x_4*x_5 + z^30*x_1*x_5^2 + z^15*x_2*x_3^2 -
2*z^15*x_2*x_3*x_4 - 2*z^30*x_2*x_3*x_5 + z^15*x_2*x_4^2 +
2*z^30*x_2*x_4*x_5 + (z^35 + z^30 - z^15 - z^10 + 1)*x_2*x_5^2
]
I3 contains a 3-dimensional subspace of CharacterRow 9
Dimension 6
Multiplicity 2
[
x_0*x_3^2 + 2*z^30*x_0*x_3*x_5 - z^30*x_0*x_4^2 + (z^35 - z^10)*x_0*x_5^2 -
z^30*x_1*x_3^2 + (-2*z^35 + 2*z^10)*x_1*x_3*x_5 + z^15*x_1*x_5^2 + (z^35
- z^10)*x_2*x_4^2,
x_0*x_3*x_4 + (z^30 + z^15)*x_0*x_3*x_5 + 1/2*(-z^30 - 1)*x_0*x_4^2 -
z^15*x_0*x_4*x_5 + 1/2*(z^35 - z^30 - z^10)*x_0*x_5^2 + 1/2*(z^35 +
2*z^30 - z^15 - z^10 + 1)*x_2*x_3^2 - z^30*x_2*x_3*x_4 + (-z^35 - z^30 +
z^15 + z^10 - 2)*x_2*x_3*x_5 + 1/2*(z^35 + z^30 - z^10)*x_2*x_4^2 +
(z^35 + z^30 - z^15 - z^10 + 1)*x_2*x_4*x_5 + 1/2*(z^35 - z^30 -
z^10)*x_2*x_5^2,
x_1*x_3*x_4 + (z^30 + z^15)*x_1*x_3*x_5 + 1/2*(z^30 - 1)*x_1*x_4^2 -
z^15*x_1*x_4*x_5 + 1/2*(z^35 - z^30 - z^10)*x_1*x_5^2 + 1/2*(z^15 +
1)*x_2*x_3^2 - x_2*x_3*x_4 + (z^35 + z^30 - 2*z^15 - z^10 +
1)*x_2*x_3*x_5 + 1/2*(z^35 + z^30 - z^15 - z^10 + 2)*x_2*x_4^2 +
z^15*x_2*x_4*x_5 + 1/2*(z^30 - 1)*x_2*x_5^2,
x_3^3 + 3*z^30*x_3^2*x_5 + (3*z^35 - 3*z^10)*x_3*x_5^2 - z^15*x_5^3,
x_3^2*x_4 + (z^30 + z^15)*x_3^2*x_5 - x_3*x_4^2 - 2*z^15*x_3*x_4*x_5 + (z^35
- z^30 - z^10)*x_3*x_5^2 + z^15*x_4^2*x_5 + z^30*x_4*x_5^2 + 1/3*(z^35 +
z^30 - 2*z^15 - z^10 + 1)*x_5^3,
x_4^3
]
I3 contains a 3-dimensional subspace of CharacterRow 11
Dimension 6
Multiplicity 2
[
x_0^3 - 3*z^30*x_0^2*x_2 + (3*z^35 - 3*z^10)*x_0*x_2^2 + z^15*x_2^3,
x_0^2*x_1 - x_0^2*x_2 - z^30*x_0*x_1^2 + z^30*x_0*x_2^2 + x_1^2*x_2 +
z^15*x_1*x_2^2 + 1/3*(-z^35 + z^30 + z^10)*x_2^3,
x_0*x_3^2 + (z^30 - z^15)*x_0*x_3*x_5 - x_0*x_4^2 + (-z^30 -
z^15)*x_0*x_4*x_5 + (-z^35 - z^30 + z^15 + z^10 - 1)*x_0*x_5^2 -
z^30*x_1*x_3^2 + (z^30 - z^15 + 1)*x_1*x_3*x_5 - x_1*x_5^2 + (-z^35 -
2*z^30 + z^15 + z^10 - 1)*x_2*x_3*x_4 + z^30*x_2*x_4^2 + (z^35 + z^30 -
z^15 - z^10 + 2)*x_2*x_4*x_5,
x_0*x_3*x_4 - x_0*x_4^2 - z^15*x_0*x_4*x_5 - z^30*x_2*x_3*x_4 +
z^30*x_2*x_4^2 + (z^35 + z^30 - z^15 - z^10 + 1)*x_2*x_4*x_5,
x_1^3 + 3*z^15*x_1^2*x_2 + 3*z^30*x_1*x_2^2 + (z^35 + z^30 - z^15 - z^10 +
1)*x_2^3,
x_1*x_3*x_4 + z^30*x_1*x_4*x_5 + z^15*x_2*x_3*x_4 + (z^35 + z^30 - z^15 -
z^10 + 1)*x_2*x_4*x_5
]
I3 contains a 6-dimensional subspace of CharacterRow 12
Dimension 12
Multiplicity 2
[
x_0^2*x_3 + 1/5*(2*z^35 - 2*z^30 - 6*z^15 - 2*z^10 + 4)*x_0*x_1*x_4 +
(-2*z^35 - 2*z^30 + 2*z^15 + 2*z^10 - 2)*x_0*x_1*x_5 -
2*z^30*x_0*x_2*x_3 + 1/5*(-2*z^35 + 2*z^30 + 6*z^15 + 2*z^10 -
4)*x_0*x_2*x_4 + (2*z^35 + 2*z^30 - 2*z^15 - 2*z^10 + 2)*x_0*x_2*x_5 -
x_1^2*x_5 + 1/5*(-2*z^35 + 2*z^30 - 4*z^15 + 2*z^10 + 6)*x_1*x_2*x_4 +
(z^35 - z^10)*x_2^2*x_3 + 1/5*(-3*z^35 - 2*z^30 + 4*z^15 + 3*z^10 -
1)*x_2^2*x_4 + x_2^2*x_5,
x_0^2*x_4 - 2*z^30*x_0*x_2*x_4 + (z^35 - z^10)*x_2^2*x_4,
x_0^2*x_5 + 1/5*(-2*z^35 + 2*z^30 - 4*z^15 + 2*z^10 - 4)*x_0*x_1*x_4 -
2*z^30*x_0*x_1*x_5 + 1/5*(2*z^35 - 2*z^30 + 4*z^15 - 2*z^10 +
4)*x_0*x_2*x_4 + (z^35 - z^10)*x_1^2*x_5 + 1/5*(-8*z^35 - 2*z^30 +
4*z^15 + 8*z^10 - 6)*x_1*x_2*x_4 + 1/5*(-2*z^35 - 3*z^30 + z^15 + 2*z^10
+ 1)*x_2^2*x_4,
x_0*x_1*x_3 - x_0*x_1*x_4 - z^15*x_0*x_1*x_5 - x_0*x_2*x_3 + x_0*x_2*x_4 +
z^15*x_0*x_2*x_5 + 1/2*(2*z^35 + z^30 - z^15 - 2*z^10 + 1)*x_1^2*x_5 +
(z^35 + z^30 - z^15 - z^10 + 1)*x_1*x_2*x_3 + (-z^35 - z^30 + z^15 +
z^10 - 1)*x_1*x_2*x_4 - x_1*x_2*x_5 + 1/2*(z^35 + z^30 - z^10)*x_2^2*x_3
+ 1/2*(-z^35 - z^30 + z^10)*x_2^2*x_4 + 1/2*(-z^35 - z^30 + z^10 -
1)*x_2^2*x_5,
x_1^2*x_3 + z^30*x_1^2*x_5 + 2*z^15*x_1*x_2*x_3 + (2*z^35 + 2*z^30 - 2*z^15
- 2*z^10 + 2)*x_1*x_2*x_5 + z^30*x_2^2*x_3 + (z^35 - z^10)*x_2^2*x_5,
x_1^2*x_4 + 2*z^15*x_1*x_2*x_4 + z^30*x_2^2*x_4,
x_3^3 + (-z^15 + 1)*x_3*x_5^2 + (2*z^30 - z^15)*x_4^2*x_5 + (3*z^35 - z^15 -
3*z^10 + 1)*x_4*x_5^2 + (-z^35 + z^10 - 1)*x_5^3,
x_3^2*x_4 + z^30*x_4^2*x_5 + (z^35 + z^30 - z^15 - z^10 + 1)*x_4*x_5^2,
x_3^2*x_5 + (z^30 - z^15)*x_3*x_5^2 - x_4^2*x_5 + (-z^30 - z^15)*x_4*x_5^2 +
(-z^35 - z^30 + z^15 + z^10 - 1)*x_5^3,
x_3*x_4^2 + z^30*x_4^2*x_5,
x_3*x_4*x_5 - 1/2*x_4^2*x_5 + 1/2*(z^30 - z^15)*x_4*x_5^2,
x_4^3 + (z^30 + z^15)*x_4^2*x_5
]
I3 contains a 12-dimensional subspace of CharacterRow 13
Dimension 18
Multiplicity 3
[
x_0^3 + (-z^35 + z^15 + z^10 - 1)*x_0*x_2^2 + (-z^30 + 2*z^15)*x_1^2*x_2 +
(-z^35 + 2*z^30 - z^15 + z^10 - 1)*x_1*x_2^2 - z^30*x_2^3,
x_0^2*x_1 + (z^30 - z^15 + 1)*x_0*x_2^2 + (-z^35 + z^10 - 1)*x_1^2*x_2 +
(-z^15 + 2)*x_1*x_2^2,
x_0^2*x_2 + (z^35 - z^15 - z^10 + 1)*x_0*x_2^2 + (-z^35 + z^10)*x_1^2*x_2 +
(z^35 - z^10 + 1)*x_1*x_2^2,
x_0^2*x_3 + z^30*x_0^2*x_5 + (z^35 - z^15 - z^10 + 1)*x_0*x_2*x_3 + (-z^35 +
z^10 - 1)*x_0*x_2*x_5 + x_2^2*x_3 + z^30*x_2^2*x_5,
x_0^2*x_4 + (z^35 - z^15 - z^10 + 1)*x_0*x_2*x_4 + (-z^35 + z^10)*x_1^2*x_4
+ (z^35 - z^10 + 1)*x_1*x_2*x_4,
x_0*x_1^2 - z^30*x_0*x_2^2 + (z^35 - z^15 - z^10 + 1)*x_1^2*x_2 + (-z^30 +
z^15 - 1)*x_1*x_2^2,
x_0*x_1*x_2 + z^15*x_0*x_2^2 - 1/2*z^30*x_1^2*x_2 + 1/2*(-z^35 - 2*z^30 +
z^15 + z^10 - 1)*x_1*x_2^2 + 1/2*(-z^35 - z^30 + z^15 + z^10 - 1)*x_2^3,
x_0*x_1*x_3 + z^15*x_0*x_2*x_3 + 1/5*(-z^35 - 4*z^30 + 3*z^15 + z^10 -
2)*x_1^2*x_4 - z^30*x_1*x_2*x_3 + 1/5*(-3*z^35 + 3*z^30 - z^15 + 3*z^10
- 1)*x_1*x_2*x_4 + (-z^35 - z^30 + z^15 + z^10 - 1)*x_2^2*x_3 +
1/5*(4*z^35 + z^30 - 2*z^15 - 4*z^10 + 3)*x_2^2*x_4,
x_0*x_1*x_4 + z^15*x_0*x_2*x_4 - z^30*x_1^2*x_4 + (-z^35 - z^30 + z^15 +
z^10 - 1)*x_1*x_2*x_4,
x_0*x_1*x_5 + z^15*x_0*x_2*x_5 + 1/5*(z^35 - z^30 + 2*z^15 - z^10 +
2)*x_1^2*x_4 + 1/5*(-2*z^35 + 2*z^30 + z^15 + 2*z^10 - 4)*x_1*x_2*x_4 -
z^30*x_1*x_2*x_5 + 1/5*(z^35 - z^30 - 3*z^15 - z^10 + 2)*x_2^2*x_4 +
(-z^35 - z^30 + z^15 + z^10 - 1)*x_2^2*x_5,
x_0*x_3^2 + (z^30 - z^15)*x_0*x_3*x_5 + (-z^35 - z^30 + z^15 + z^10 -
1)*x_0*x_5^2 - z^30*x_2*x_3^2 + (z^30 - z^15 + 1)*x_2*x_3*x_5 -
x_2*x_5^2,
x_0*x_3*x_4 + z^30*x_0*x_4*x_5 - z^30*x_2*x_3*x_4 + (-z^35 +
z^10)*x_2*x_4*x_5,
x_0*x_4^2 + (z^30 + z^15)*x_0*x_4*x_5 + (-z^35 - 2*z^30 + z^15 + z^10 -
1)*x_2*x_3*x_4 + (z^35 + z^30 - z^15 - z^10 + 1)*x_2*x_4^2,
x_1^3 + (2*z^15 - 1)*x_1^2*x_2 + (z^30 - 2*z^15)*x_1*x_2^2 - z^30*x_2^3,
x_1^2*x_3 - x_1^2*x_4 - z^15*x_1^2*x_5 + (z^15 - 1)*x_1*x_2*x_3 + (-z^15 +
1)*x_1*x_2*x_4 + (-z^30 + z^15)*x_1*x_2*x_5 - z^15*x_2^2*x_3 +
z^15*x_2^2*x_4 + z^30*x_2^2*x_5,
x_1*x_3^2 + (z^30 - z^15)*x_1*x_3*x_5 + (-z^35 - z^30 + z^15 + z^10 -
1)*x_1*x_5^2 + z^15*x_2*x_3^2 + (-z^15 - 1)*x_2*x_3*x_4 + (z^35 - z^15 -
z^10 + 1)*x_2*x_3*x_5 + (-z^35 - 2*z^30 + z^15 + z^10 - 1)*x_2*x_4*x_5 +
(-z^35 + z^10)*x_2*x_5^2,
x_1*x_3*x_4 + z^30*x_1*x_4*x_5 - x_2*x_3*x_4 - z^30*x_2*x_4*x_5,
x_1*x_4^2 + (z^30 + z^15)*x_1*x_4*x_5 + (-z^15 - 1)*x_2*x_3*x_4 +
z^15*x_2*x_4^2
]
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 the file autcv10e.txt to analyze this 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);
> PlaneCharacters(T,T[4]+T[8]);
[
( 3, 1, 0, zeta(5)_5^3 + 2*zeta(5)_5, zeta(5)_5^3 - zeta(5)_5^2 - zeta(5)_5
- 1, 2*zeta(5)_5^2 + zeta(5)_5, -2*zeta(5)_5^3 - zeta(5)_5^2 -
2*zeta(5)_5 - 2, -zeta(5)_5^3 - zeta(5)_5^2, zeta(5)_5^3 + zeta(5)_5^2 +
1, zeta(5)_5^2, -zeta(5)_5^3 - zeta(5)_5^2 - zeta(5)_5 - 1, zeta(5)_5,
zeta(5)_5^3 ),
( 3, -1, 0, zeta(5)_5^3 + 2*zeta(5)_5, zeta(5)_5^3 - zeta(5)_5^2 - zeta(5)_5
- 1, 2*zeta(5)_5^2 + zeta(5)_5, -2*zeta(5)_5^3 - zeta(5)_5^2 -
2*zeta(5)_5 - 2, -zeta(5)_5^3 - zeta(5)_5^2, zeta(5)_5^3 + zeta(5)_5^2 +
1, -zeta(5)_5^2, zeta(5)_5^3 + zeta(5)_5^2 + zeta(5)_5 + 1, -zeta(5)_5,
-zeta(5)_5^3 )
]
> psis:=PlaneCharacters(T,T[4]+T[8]);
> psis[1] eq T[7];
true
> psis[2] eq T[10];
true
We see that two degree three characters satisfy this description. They are the irreducible characters \( \chi_7\) and \( \chi_{10}\) in the character table above.
Next, for each of these characters \( \psi \), 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,
0,
1,
1,
0,
0,
0,
1,
0,
0,
0,
1,
1
]
( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
> Decomposition(T,Symmetrization(psis[2],[5]));
[
0,
1,
1,
0,
0,
1,
0,
0,
1,
0,
0,
1,
1
]
( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 )
This shows that the quintic could either come from the isotypical subspace of \(S_5\) with character \( \chi_1\) if \(G\) acts with character \( \chi_{7}\) on \( \mathbb{P}^2\), or from the isotypical subspace of \(S_5\) with character \( \chi_2\) if \(G\) acts with character \( \chi_{10}\) on \( \mathbb{P}^2\).
We compute the quintic in three variables with character \(\chi_1\) when \(G\) acts with character \(\chi_7\).
> G7:=ActionGenerators(GModule(T[7]));
Cyclotomic Field of order 5 and degree 4
> G7;
[
[ -z_5^2 -z_5^2 - 1 z_5^3 + z_5^2 + 1]
[ -1 0 z_5^3 + z_5^2 + z_5 + 1]
[ z_5^3 z_5^3 z_5^2 + 1],
[ -z_5^2 -z_5^2 z_5^3 + z_5^2 + 1]
[ -1 -1 z_5^3 + z_5^2 + z_5 + 1]
[ z_5^3 z_5^3 + z_5 z_5^2 + 1],
[ z_5^2 0 0]
[ 0 -z_5^3 - 1 z_5^3 + z_5]
[ 0 -z_5^2 - 1 z_5^3 + z_5^2 + z_5 + 1],
[ 1 1 -z_5^3 - z_5^2 - z_5 - 1]
[ 0 z_5^3 + z_5^2 + 1 -z_5^3 - z_5 - 1]
[ 0 -z_5^3 - z_5 -1]
> K<z_5>:=CyclotomicField(5);
> GL3K:=GeneralLinearGroup(3,K);
> rho7:=homGL3K | G7>;
> rho7(G.1 * G.3 * G.4^2);
[z_5^3 + z_5^2 + z_5 + 1 z_5 1]
[ -z_5^2 -z_5^2 z_5^3 + z_5^2 + 1]
[ 1 z_5^3 + 1 -z_5^3 - z_5]
> rho7(G.2^2 * G.3^4 * G.4^4);
[ 0 z_5^3 + z_5 z_5^3 + z_5^2 + 1]
[ z_5^3 -z_5^2 - 1 z_5^3 + z_5^2 + z_5 + 1]
[z_5^3 + z_5^2 + z_5 + 1 z_5^3 + z_5 + 1 z_5^2 + 1]
> S<y_0,y_1,y_2>:=PolynomialRing(K,3);
> f5:=IsotypicalSubspace(G,rho7,T,S,5,1);
CharacterRow 1
Dimension 1
Multiplicity 1
> f5;
[
y_0^4*y_1 + 1/5*(3*z_5^3 + z_5^2 - z_5 + 2)*y_0^4*y_2 + (2*z_5^3 + 2*z_5 +
2)*y_0^3*y_1^2 + 1/5*(8*z_5^3 - 4*z_5^2 + 4*z_5 - 8)*y_0^3*y_1*y_2 +
1/5*(-6*z_5^3 - 2*z_5^2 + 2*z_5 - 4)*y_0^3*y_2^2 + (-2*z_5^2 -
2)*y_0^2*y_1^3 + 1/5*(-12*z_5^3 + 6*z_5^2 - 6*z_5 - 18)*y_0^2*y_1^2*y_2
+ 1/5*(-12*z_5^3 + 6*z_5^2 - 6*z_5 + 12)*y_0^2*y_1*y_2^2 + 1/5*(6*z_5^3
+ 2*z_5^2 - 2*z_5 + 4)*y_0^2*y_2^3 + (-z_5^3 - z_5^2 - z_5 -
1)*y_0*y_1^4 + 1/5*(8*z_5^3 + 16*z_5^2 + 4*z_5 + 12)*y_0*y_1^3*y_2 +
1/5*(12*z_5^3 - 6*z_5^2 + 6*z_5 + 18)*y_0*y_1^2*y_2^2 + 1/5*(8*z_5^3 -
4*z_5^2 + 4*z_5 - 8)*y_0*y_1*y_2^3 + 1/5*(-3*z_5^3 - z_5^2 + z_5 -
2)*y_0*y_2^4 + 1/5*(3*z_5^3 + z_5^2 + 4*z_5 + 2)*y_1^4*y_2 +
1/5*(-4*z_5^3 - 8*z_5^2 - 2*z_5 - 6)*y_1^3*y_2^2 + 1/5*(-4*z_5^3 +
2*z_5^2 - 2*z_5 - 6)*y_1^2*y_2^3 + 1/5*(-2*z_5^3 + z_5^2 - z_5 +
2)*y_1*y_2^4
]
In the next section we check that this polynomial defines a smooth plane quintic with the correct automorphisms..
> K<z_5>:=CyclotomicField(5);
> P2<y_0,y_1,y_2>:=ProjectiveSpace(K,2);
> X:=Scheme(P2,[
> y_0^4*y_1 - z_5^3*y_0^4*y_2 + 2*z_5^3*y_0^3*y_1^2 - 4*z_5*y_0^3*y_1*y_2 + (-\
2*z_5^3 - 2*z_5^2 - 2*z_5 - 2)*y_0^3*y_2^2 + 2*z_5*y_0^2*y_1^3 + (6*z_5^3 + 6*\
z_5^2 + 6*z_5 + 6)*y_0^2*y_1^2*y_2 + 6*z_5^2*y_0^2*y_1*y_2^2 - 2*y_0^2*y_2^3 +\
(-z_5^3 - z_5^2 - z_5 - 1)*y_0*y_1^4 - 4*z_5^2*y_0*y_1^3*y_2+ 6*y_0*y_1^2*y_2\
^2 - 4*z_5^3*y_0*y_1*y_2^3 + z_5*y_0*y_2^4 + (z_5^3 - 1)*y_1^4*y_2 + (4*z_5^3 \
+ 2*z_5^2 + 2*z_5 + 2)*y_1^3*y_2^2 + (-2*z_5 + 2)*y_1^2*y_2^3 + (-z_5^3 - z_5^\
2 - 2*z_5 - 1)*y_1*y_2^4]);
> Dimension(X);
1
> IsSingular(X);
false
> A:=Matrix([
> [z_5^3 + z_5^2 + z_5 + 1, z_5, 1],
> [-z_5^2, -z_5^2, z_5^3 + z_5^2 + 1],
> [1, z_5^3 + 1, -z_5^3 - z_5]
> ]);
> B:=Matrix([
> [0, z_5^3 + z_5, z_5^3 + z_5^2 + 1],
> [z_5^3, -z_5^2 - 1, z_5^3 + z_5^2 + z_5 + 1],
> [z_5^3 + z_5^2 + z_5 + 1, z_5^3 + z_5 + 1, z_5^2 + 1]
> ]);
> GL3K:=GeneralLinearGroup(3,K);
> Order(A);
2
> Order(B);
3
> Order( (A*B)^(-1));
10
> IdentifyGroup(sub<GL3K | A,B>);
<150, 5>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
(z_5^3 + z_5^2 + z_5 + 1)*y_0 - z_5^2*y_1 + y_2
z_5*y_0 - z_5^2*y_1 + (z_5^3 + 1)*y_2
y_0 + (z_5^3 + z_5^2 + 1)*y_1 + (-z_5^3 - z_5)*y_2
and inverse
(z_5^3 + z_5^2 + z_5 + 1)*y_0 - z_5^2*y_1 + y_2
z_5*y_0 - z_5^2*y_1 + (z_5^3 + 1)*y_2
y_0 + (z_5^3 + z_5^2 + 1)*y_1 + (-z_5^3 - z_5)*y_2
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
z_5^3*y_1 + (z_5^3 + z_5^2 + z_5 + 1)*y_2
(z_5^3 + z_5)*y_0 + (-z_5^2 - 1)*y_1 + (z_5^3 + z_5 + 1)*y_2
(z_5^3 + z_5^2 + 1)*y_0 + (z_5^3 + z_5^2 + z_5 + 1)*y_1 + (z_5^2 + 1)*y_2
and inverse
-z_5^2*y_0 - y_1 + z_5^3*y_2
(-z_5^3 - 1)*y_1 + (-z_5^2 - 1)*y_2
-z_5*y_0 + (z_5^3 + z_5 + 1)*y_1 + (z_5^3 + z_5^2 + 1)*y_2
> K<z_5>:=CyclotomicField(5);
> M1:=Matrix([
> [z_5,0,0],
> [0,z_5^4,0],
> [0,0,1]
> ]);
> M2:=Matrix([
> [0,-1,0],
> [-1,0,0],
> [0,0,-1]
> ]);
> M3:=Matrix([
> [0,1,0],
> [0,0,1],
> [1,0,0]
> ]);
> GL3K:=GeneralLinearGroup(3,K);
> G:=SmallGroup(150,5);
> b,f:=IsIsomorphic(G,sub<GL3K | M1,M2,M3>);
> b;
true
> f;
Mapping from: GrpPC: G to MatrixGroup(3, K) of order 2 * 3 * 5^2
Composition of Mapping from: GrpPC: G to GrpPC and
Mapping from: GrpPC to MatrixGroup(3, K) of order 2 * 3 * 5^2
> A:=f(G.1 * G.3 * G.4^2);
> B:=f(G.2^2 * G.3^4 * G.4^4);
> A;
[ 0 -z_5^3 0]
[-z_5^2 0 0]
[ 0 0 -1]
> B;
[ 0 0 z_5^3]
[ 1 0 0]
[ 0 z_5^2 0]
> P2<y_0,y_1,y_2>:=ProjectiveSpace(K,2);
> X:=Scheme(P2,[y_0^5+y_1^5+y_2^5]);
> Dimension(X);
1
> IsSingular(X);
false
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
-z_5^2*y_1
-z_5^3*y_0
-y_2
and inverse
-z_5^2*y_1
-z_5^3*y_0
-y_2
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
y_1
z_5^2*y_2
z_5^3*y_0
and inverse
z_5^2*y_2
y_0
z_5^3*y_1