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Magaard, Shaska, Shpectorov, and Völklein give tables of 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 7 Riemann surface with automorphism group (504,156) 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 (2,3,7). This curve is well-known in the literature as Macbeath's curve.
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 V2.21-4 Mon Sep 14 2015 08:59:18 on ace-math01 [Seed = 3964632531]
Type ? for help. Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> G:=SmallGroup(504,156);
> G;
Permutation group G acting on a set of cardinality 9
(1, 9, 8, 6, 5, 4, 2)
(1, 2, 8, 9, 7, 5, 3)
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,7,[2,3,7]);
Set seed to 0.
Character Table of Group G
--------------------------
-----------------------------------------------
Class | 1 2 3 4 5 6 7 8 9
Size | 1 63 56 72 72 72 56 56 56
Order | 1 2 3 7 7 7 9 9 9
-----------------------------------------------
p = 2 1 1 3 5 6 4 8 9 7
p = 3 1 2 1 6 4 5 3 3 3
p = 7 1 2 3 1 1 1 8 9 7
-----------------------------------------------
X.1 + 1 1 1 1 1 1 1 1 1
X.2 + 7 -1 -2 0 0 0 1 1 1
X.3 + 7 -1 1 0 0 0 Z2 Z2#2 Z2#4
X.4 + 7 -1 1 0 0 0 Z2#4 Z2 Z2#2
X.5 + 7 -1 1 0 0 0 Z2#2 Z2#4 Z2
X.6 + 8 0 -1 1 1 1 -1 -1 -1
X.7 + 9 1 0 Z1 Z1#2 Z1#3 0 0 0
X.8 + 9 1 0 Z1#3 Z1 Z1#2 0 0 0
X.9 + 9 1 0 Z1#2 Z1#3 Z1 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(7: Sparse := true)) ! [ RationalField() | 0, 0, 0, 1,
1, 0 ]
Z2 = (CyclotomicField(9: Sparse := true)) ! [ RationalField() | 0, 0, 0, 0,
-1, -1 ]
Conjugacy Classes of group G
----------------------------
[1] Order 1 Length 1
Rep Id(G)
[2] Order 2 Length 63
Rep (2, 6)(3, 4)(5, 9)(7, 8)
[3] Order 3 Length 56
Rep (1, 2, 5)(3, 9, 7)(4, 6, 8)
[4] Order 7 Length 72
Rep (1, 2, 8, 9, 7, 5, 3)
[5] Order 7 Length 72
Rep (1, 8, 7, 3, 2, 9, 5)
[6] Order 7 Length 72
Rep (1, 9, 3, 8, 5, 2, 7)
[7] Order 9 Length 56
Rep (1, 3, 4, 2, 9, 6, 5, 7, 8)
[8] Order 9 Length 56
Rep (1, 4, 9, 5, 8, 3, 2, 6, 7)
[9] Order 9 Length 56
Rep (1, 9, 8, 2, 7, 4, 5, 3, 6)
Surface kernel generators: [
(1, 2)(4, 8)(5, 7)(6, 9),
(1, 6, 9)(2, 7, 3)(4, 5, 8),
(1, 6, 2, 3, 5, 8, 7)
]
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 ]
S_1 =[ 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
H^0(C,K) =[ 0, 1, 0, 0, 0, 0, 0, 0, 0 ]
I_2 =[ 1, 0, 0, 0, 0, 0, 0, 0, 1 ]
S_2 =[ 1, 0, 0, 0, 0, 0, 1, 1, 1 ]
H^0(C,2K)=[ 0, 0, 0, 0, 0, 0, 1, 1, 0 ]
I_3 =[ 0, 1, 1, 1, 1, 1, 0, 1, 1 ]
S_3 =[ 0, 1, 2, 2, 2, 1, 1, 1, 1 ]
H^0(C,3K)=[ 0, 0, 1, 1, 1, 0, 1, 0, 0 ]
I2timesS1=[ 0, 2, 1, 1, 1, 1, 1, 1, 1 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 504 and degree 144
[
[ 0 -1 0 0 0 0 0]
[-1/2 1 1 -1/2 1 1/2 -1/2]
[ 1/2 -1 0 1/2 -1 -1/2 1/2]
[ 1 -1 -1 0 0 0 -1]
[ 1/2 -1 -1 1/2 0 -1/2 -1/2]
[ 0 -1 0 0 -1 -1 0]
[ 0 -1 -1 1 -1 0 0],
[ -1 1 1 0 0 0 1]
[ 0 -1 0 1 -1 0 0]
[-1/2 2 1 -1/2 1 1/2 1/2]
[-1/2 0 0 -1/2 1 1/2 -1/2]
[-1/2 0 0 -1/2 1 -1/2 -1/2]
[ 1/2 1 0 -1/2 1 1/2 1/2]
[ 0 0 -1 0 0 0 0]
]
Matrix Surface Kernel Generators:
[
[-1/2 1 0 -1/2 1 1/2 -1/2]
[ 1/2 1 0 -1/2 0 1/2 1/2]
[-1/2 0 0 -1/2 1 -1/2 -1/2]
[ 1/2 -1 -1 -1/2 0 1/2 -1/2]
[ 0 -1 0 0 0 0 -1]
[ 1/2 -1 -1 1/2 0 -1/2 -1/2]
[-1/2 -1 0 1/2 -1 -1/2 -1/2],
[-1/2 0 0 -1/2 0 -1/2 1/2]
[ 1/2 -1 -1 1/2 0 -1/2 -1/2]
[ 0 0 0 0 -1 0 1]
[ 1/2 -1 0 -1/2 0 -1/2 -1/2]
[-1/2 0 1 -1/2 0 -1/2 -1/2]
[ 0 1 1 0 0 0 1]
[ -1 0 1 0 0 0 0],
[ 1 -1 -1 0 0 0 0]
[ 0 1 1 0 1 0 0]
[ 1/2 -2 -1 1/2 -1 -1/2 -1/2]
[ 1/2 0 0 1/2 -1 -1/2 1/2]
[ 1/2 0 -1 1/2 -1 1/2 1/2]
[-1/2 -1 0 1/2 -1 -1/2 1/2]
[ 0 0 -1 0 0 0 0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
x_0^2 + x_0*x_2 + x_1^2 - x_1*x_2 - x_1*x_4 - x_1*x_5 - x_1*x_6 + x_2^2 +
x_2*x_5 + x_3^2 + x_3*x_4 + x_4^2 + x_4*x_6 + x_5^2 + x_6^2
]
I2 contains a 9-dimensional subspace of CharacterRow 9
Dimension 9
Multiplicity 1
[
x_0^2 + 1/2*(3*z^132 - 3*z^48 - 3*z^36 - 2)*x_1*x_3 + 1/2*(z^132 - z^96 +
z^72 - z^48 - z^36 + z^12 - 1)*x_1*x_4 + 1/2*(2*z^132 + z^96 - z^72 -
2*z^48 - 2*z^36 - z^12 + 1)*x_1*x_5 + 1/2*(3*z^132 - z^96 + z^72 -
3*z^48 - 3*z^36 + z^12 + 1)*x_1*x_6 + 1/2*(-z^132 - 2*z^96 + 2*z^72 +
z^48 + z^36 + 2*z^12 + 3)*x_2*x_3 + 1/2*(-z^132 + z^48 + z^36)*x_2*x_4 -
1/2*x_2*x_5 + 1/2*(-3*z^132 + 2*z^96 - 2*z^72 + 3*z^48 + 3*z^36 - 2*z^12
- 2)*x_2*x_6 + 1/2*(z^132 + z^96 - z^72 - z^48 - z^36 - z^12 - 2)*x_3^2
+ 1/2*(-z^132 + 2*z^96 - 2*z^72 + z^48 + z^36 - 2*z^12 - 2)*x_3*x_4 +
1/2*(-z^132 - z^96 + z^72 + z^48 + z^36 + z^12)*x_3*x_5 + 1/2*(-3*z^132
+ z^96 - z^72 + 3*z^48 + 3*z^36 - z^12 + 2)*x_3*x_6 + 1/2*(z^96 - z^72 -
z^12)*x_4^2 + 1/2*(-2*z^132 + 2*z^96 - 2*z^72 + 2*z^48 + 2*z^36 - 2*z^12
- 3)*x_4*x_5 + 1/2*(-z^132 + z^48 + z^36 + 3)*x_4*x_6 + 1/2*(-z^132 -
z^96 + z^72 + z^48 + z^36 + z^12 - 1)*x_5^2 + 1/2*(-3*z^132 + z^96 -
z^72 + 3*z^48 + 3*z^36 - z^12 - 2)*x_5*x_6 + 1/2*(-z^132 - z^96 + z^72 +
z^48 + z^36 + z^12 + 1)*x_6^2,
x_0*x_1 + 1/2*(-5*z^132 + 5*z^96 - 5*z^72 + 5*z^48 + 5*z^36 - 5*z^12 +
10)*x_1*x_3 + 1/2*(-16*z^132 + 27*z^96 - 27*z^72 + 16*z^48 + 16*z^36 -
27*z^12 + 8)*x_1*x_4 + 1/2*(11*z^132 - 26*z^96 + 26*z^72 - 11*z^48 -
11*z^36 + 26*z^12 - 8)*x_1*x_5 + 1/2*(-22*z^132 + 37*z^96 - 37*z^72 +
22*z^48 + 22*z^36 - 37*z^12 + 8)*x_1*x_6 + (2*z^132 - 4*z^96 + 4*z^72 -
2*z^48 - 2*z^36 + 4*z^12 - 2)*x_2^2 + 1/2*(-2*z^132 + 7*z^96 - 7*z^72 +
2*z^48 + 2*z^36 - 7*z^12 - 5)*x_2*x_3 + 1/2*(z^132 + z^96 - z^72 - z^48
- z^36 - z^12 + 2)*x_2*x_4 + 1/2*(-z^132 + z^48 + z^36 - 1)*x_2*x_5 +
1/2*(23*z^132 - 39*z^96 + 39*z^72 - 23*z^48 - 23*z^36 + 39*z^12 -
6)*x_2*x_6 + 1/2*(9*z^132 - 17*z^96 + 17*z^72 - 9*z^48 - 9*z^36 +
17*z^12 + 1)*x_3^2 + 1/2*(5*z^132 - 11*z^96 + 11*z^72 - 5*z^48 - 5*z^36
+ 11*z^12 - 4)*x_3*x_4 + 1/2*(-19*z^132 + 35*z^96 - 35*z^72 + 19*z^48 +
19*z^36 - 35*z^12 + 5)*x_3*x_5 + 1/2*(5*z^132 - 3*z^96 + 3*z^72 - 5*z^48
- 5*z^36 + 3*z^12 - 7)*x_3*x_6 + 1/2*(6*z^132 - 14*z^96 + 14*z^72 -
6*z^48 - 6*z^36 + 14*z^12 - 5)*x_4^2 + 1/2*(-15*z^132 + 30*z^96 -
30*z^72 + 15*z^48 + 15*z^36 - 30*z^12 + 13)*x_4*x_5 + 1/2*(16*z^132 -
23*z^96 + 23*z^72 - 16*z^48 - 16*z^36 + 23*z^12 - 7)*x_4*x_6 +
1/2*(-6*z^132 + 13*z^96 - 13*z^72 + 6*z^48 + 6*z^36 - 13*z^12 + 2)*x_5^2
+ 1/2*(9*z^132 - 15*z^96 + 15*z^72 - 9*z^48 - 9*z^36 + 15*z^12 -
3)*x_5*x_6 + 1/2*(2*z^132 + 3*z^96 - 3*z^72 - 2*z^48 - 2*z^36 - 3*z^12 +
2)*x_6^2,
x_0*x_2 + 1/2*(-9*z^132 + 8*z^96 - 8*z^72 + 9*z^48 + 9*z^36 - 8*z^12 +
6)*x_1*x_3 + 1/2*(-17*z^132 + 29*z^96 - 29*z^72 + 17*z^48 + 17*z^36 -
29*z^12 + 9)*x_1*x_4 + 1/2*(12*z^132 - 27*z^96 + 27*z^72 - 12*z^48 -
12*z^36 + 27*z^12 - 7)*x_1*x_5 + 1/2*(-23*z^132 + 39*z^96 - 39*z^72 +
23*z^48 + 23*z^36 - 39*z^12 + 9)*x_1*x_6 + (3*z^132 - 5*z^96 + 5*z^72 -
3*z^48 - 3*z^36 + 5*z^12 - 1)*x_2^2 + 1/2*(-z^132 + 6*z^96 - 6*z^72 +
z^48 + z^36 - 6*z^12 - 3)*x_2*x_3 + 1/2*(-z^132 + 2*z^96 - 2*z^72 + z^48
+ z^36 - 2*z^12)*x_2*x_4 + 1/2*(2*z^132 - 2*z^96 + 2*z^72 - 2*z^48 -
2*z^36 + 2*z^12 + 1)*x_2*x_5 + 1/2*(23*z^132 - 40*z^96 + 40*z^72 -
23*z^48 - 23*z^36 + 40*z^12 - 8)*x_2*x_6 + 1/2*(7*z^132 - 17*z^96 +
17*z^72 - 7*z^48 - 7*z^36 + 17*z^12 - 2)*x_3^2 + 1/2*(7*z^132 - 12*z^96
+ 12*z^72 - 7*z^48 - 7*z^36 + 12*z^12 - 2)*x_3*x_4 + 1/2*(-19*z^132 +
37*z^96 - 37*z^72 + 19*z^48 + 19*z^36 - 37*z^12 + 8)*x_3*x_5 +
1/2*(7*z^132 - 5*z^96 + 5*z^72 - 7*z^48 - 7*z^36 + 5*z^12 - 6)*x_3*x_6 +
1/2*(8*z^132 - 15*z^96 + 15*z^72 - 8*z^48 - 8*z^36 + 15*z^12 - 4)*x_4^2
+ 1/2*(-18*z^132 + 32*z^96 - 32*z^72 + 18*z^48 + 18*z^36 - 32*z^12 +
11)*x_4*x_5 + 1/2*(15*z^132 - 24*z^96 + 24*z^72 - 15*z^48 - 15*z^36 +
24*z^12 - 11)*x_4*x_6 + 1/2*(-5*z^132 + 13*z^96 - 13*z^72 + 5*z^48 +
5*z^36 - 13*z^12 + 3)*x_5^2 + 1/2*(11*z^132 - 17*z^96 + 17*z^72 -
11*z^48 - 11*z^36 + 17*z^12 - 2)*x_5*x_6 + 1/2*(-z^132 + 5*z^96 - 5*z^72
+ z^48 + z^36 - 5*z^12 - 1)*x_6^2,
x_0*x_3 + 1/4*(z^132 - 2*z^96 + 2*z^72 - z^48 - z^36 + 2*z^12 - 2)*x_1*x_3 +
1/4*(5*z^132 - 11*z^96 + 11*z^72 - 5*z^48 - 5*z^36 + 11*z^12 -
5)*x_1*x_4 + 1/4*(-4*z^132 + 13*z^96 - 13*z^72 + 4*z^48 + 4*z^36 -
13*z^12 + 5)*x_1*x_5 + 1/4*(11*z^132 - 19*z^96 + 19*z^72 - 11*z^48 -
11*z^36 + 19*z^12 - 3)*x_1*x_6 + 1/2*(-z^132 + 2*z^96 - 2*z^72 + z^48 +
z^36 - 2*z^12 + 1)*x_2^2 + 1/4*(3*z^132 - 4*z^96 + 4*z^72 - 3*z^48 -
3*z^36 + 4*z^12 + 5)*x_2*x_3 + 1/4*(z^132 - 2*z^96 + 2*z^72 - z^48 -
z^36 + 2*z^12)*x_2*x_4 + 1/4*(-2*z^132 + 2*z^48 + 2*z^36 + 1)*x_2*x_5 +
1/4*(-13*z^132 + 20*z^96 - 20*z^72 + 13*z^48 + 13*z^36 - 20*z^12 +
2)*x_2*x_6 + 1/4*(-5*z^132 + 7*z^96 - 7*z^72 + 5*z^48 + 5*z^36 - 7*z^12
+ 2)*x_3^2 + 1/4*(-3*z^132 + 4*z^96 - 4*z^72 + 3*z^48 + 3*z^36 - 4*z^12
+ 2)*x_3*x_4 + 1/4*(9*z^132 - 19*z^96 + 19*z^72 - 9*z^48 - 9*z^36 +
19*z^12 - 2)*x_3*x_5 + 1/4*(-z^132 + 3*z^96 - 3*z^72 + z^48 + z^36 -
3*z^12)*x_3*x_6 + 1/4*(-2*z^132 + 5*z^96 - 5*z^72 + 2*z^48 + 2*z^36 -
5*z^12 + 4)*x_4^2 + 1/4*(8*z^132 - 18*z^96 + 18*z^72 - 8*z^48 - 8*z^36 +
18*z^12 - 5)*x_4*x_5 + 1/4*(-5*z^132 + 10*z^96 - 10*z^72 + 5*z^48 +
5*z^36 - 10*z^12 + 5)*x_4*x_6 + 1/4*(z^132 - 5*z^96 + 5*z^72 - z^48 -
z^36 + 5*z^12 - 3)*x_5^2 + 1/4*(-5*z^132 + 7*z^96 - 7*z^72 + 5*z^48 +
5*z^36 - 7*z^12)*x_5*x_6 + 1/4*(z^132 - z^96 + z^72 - z^48 - z^36 + z^12
- 1)*x_6^2,
x_0*x_4 + 1/2*(-9*z^132 + 12*z^96 - 12*z^72 + 9*z^48 + 9*z^36 - 12*z^12 +
7)*x_1*x_3 + 1/2*(-23*z^132 + 42*z^96 - 42*z^72 + 23*z^48 + 23*z^36 -
42*z^12 + 12)*x_1*x_4 + 1/2*(20*z^132 - 40*z^96 + 40*z^72 - 20*z^48 -
20*z^36 + 40*z^12 - 9)*x_1*x_5 + 1/2*(-31*z^132 + 56*z^96 - 56*z^72 +
31*z^48 + 31*z^36 - 56*z^12 + 14)*x_1*x_6 + (4*z^132 - 7*z^96 + 7*z^72 -
4*z^48 - 4*z^36 + 7*z^12 - 2)*x_2^2 + 1/2*(-2*z^132 + 7*z^96 - 7*z^72 +
2*z^48 + 2*z^36 - 7*z^12 - 2)*x_2*x_3 + 1/2*(-z^132 + 2*z^96 - 2*z^72 +
z^48 + z^36 - 2*z^12 + 1)*x_2*x_4 + 1/2*(z^132 - z^96 + z^72 - z^48 -
z^36 + z^12 - 1)*x_2*x_5 + 1/2*(33*z^132 - 58*z^96 + 58*z^72 - 33*z^48 -
33*z^36 + 58*z^12 - 13)*x_2*x_6 + 1/2*(12*z^132 - 23*z^96 + 23*z^72 -
12*z^48 - 12*z^36 + 23*z^12 - 5)*x_3^2 + 1/2*(9*z^132 - 16*z^96 +
16*z^72 - 9*z^48 - 9*z^36 + 16*z^12 - 5)*x_3*x_4 + 1/2*(-28*z^132 +
53*z^96 - 53*z^72 + 28*z^48 + 28*z^36 - 53*z^12 + 11)*x_3*x_5 +
1/2*(6*z^132 - 7*z^96 + 7*z^72 - 6*z^48 - 6*z^36 + 7*z^12 - 5)*x_3*x_6 +
1/2*(11*z^132 - 21*z^96 + 21*z^72 - 11*z^48 - 11*z^36 + 21*z^12 -
6)*x_4^2 + 1/2*(-27*z^132 + 49*z^96 - 49*z^72 + 27*z^48 + 27*z^36 -
49*z^12 + 13)*x_4*x_5 + 1/2*(20*z^132 - 35*z^96 + 35*z^72 - 20*z^48 -
20*z^36 + 35*z^12 - 12)*x_4*x_6 + 1/2*(-9*z^132 + 18*z^96 - 18*z^72 +
9*z^48 + 9*z^36 - 18*z^12 + 4)*x_5^2 + 1/2*(14*z^132 - 23*z^96 + 23*z^72
- 14*z^48 - 14*z^36 + 23*z^12 - 5)*x_5*x_6 + 1/2*(-3*z^132 + 6*z^96 -
6*z^72 + 3*z^48 + 3*z^36 - 6*z^12)*x_6^2,
x_0*x_5 + 1/4*(-11*z^132 + 22*z^96 - 22*z^72 + 11*z^48 + 11*z^36 - 22*z^12 +
4)*x_1*x_3 + 1/4*(-35*z^132 + 63*z^96 - 63*z^72 + 35*z^48 + 35*z^36 -
63*z^12 + 13)*x_1*x_4 + 1/4*(32*z^132 - 57*z^96 + 57*z^72 - 32*z^48 -
32*z^36 + 57*z^12 - 15)*x_1*x_5 + 1/4*(-45*z^132 + 83*z^96 - 83*z^72 +
45*z^48 + 45*z^36 - 83*z^12 + 19)*x_1*x_6 + 1/2*(5*z^132 - 10*z^96 +
10*z^72 - 5*z^48 - 5*z^36 + 10*z^12 - 3)*x_2^2 + 1/4*(-3*z^132 + 6*z^96
- 6*z^72 + 3*z^48 + 3*z^36 - 6*z^12 + 3)*x_2*x_3 + 1/4*(z^132 + 2*z^96 -
2*z^72 - z^48 - z^36 - 2*z^12 + 2)*x_2*x_4 + 1/4*(-2*z^96 + 2*z^72 +
2*z^12 + 1)*x_2*x_5 + 1/4*(47*z^132 - 84*z^96 + 84*z^72 - 47*z^48 -
47*z^36 + 84*z^12 - 20)*x_2*x_6 + 1/4*(21*z^132 - 33*z^96 + 33*z^72 -
21*z^48 - 21*z^36 + 33*z^12 - 8)*x_3^2 + 1/4*(13*z^132 - 24*z^96 +
24*z^72 - 13*z^48 - 13*z^36 + 24*z^12 - 8)*x_3*x_4 + 1/4*(-41*z^132 +
73*z^96 - 73*z^72 + 41*z^48 + 41*z^36 - 73*z^12 + 20)*x_3*x_5 +
1/4*(5*z^132 - 13*z^96 + 13*z^72 - 5*z^48 - 5*z^36 + 13*z^12 -
2)*x_3*x_6 + 1/4*(16*z^132 - 31*z^96 + 31*z^72 - 16*z^48 - 16*z^36 +
31*z^12 - 8)*x_4^2 + 1/4*(-38*z^132 + 72*z^96 - 72*z^72 + 38*z^48 +
38*z^36 - 72*z^12 + 19)*x_4*x_5 + 1/4*(29*z^132 - 56*z^96 + 56*z^72 -
29*z^48 - 29*z^36 + 56*z^12 - 9)*x_4*x_6 + 1/4*(-15*z^132 + 25*z^96 -
25*z^72 + 15*z^48 + 15*z^36 - 25*z^12 + 7)*x_5^2 + 1/4*(17*z^132 -
33*z^96 + 33*z^72 - 17*z^48 - 17*z^36 + 33*z^12 - 10)*x_5*x_6 +
1/4*(-3*z^132 + 5*z^96 - 5*z^72 + 3*z^48 + 3*z^36 - 5*z^12 + 5)*x_6^2,
x_0*x_6 + 1/4*(5*z^132 + 4*z^96 - 4*z^72 - 5*z^48 - 5*z^36 - 4*z^12)*x_1*x_3
+ 1/4*(-z^132 + 3*z^96 - 3*z^72 + z^48 + z^36 - 3*z^12 - 1)*x_1*x_4 +
1/4*(-2*z^132 + z^96 - z^72 + 2*z^48 + 2*z^36 - z^12 - 5)*x_1*x_5 +
1/4*(-3*z^132 + 3*z^96 - 3*z^72 + 3*z^48 + 3*z^36 - 3*z^12 - 3)*x_1*x_6
+ 1/2*(-z^132 + z^48 + z^36 - 1)*x_2^2 + 1/4*(-5*z^132 + 5*z^48 + 5*z^36
- 1)*x_2*x_3 + 1/4*(z^132 - z^48 - z^36 + 2)*x_2*x_4 + 1/4*(-2*z^132 +
2*z^96 - 2*z^72 + 2*z^48 + 2*z^36 - 2*z^12 - 3)*x_2*x_5 + 1/4*(3*z^132 -
2*z^96 + 2*z^72 - 3*z^48 - 3*z^36 + 2*z^12 + 4)*x_2*x_6 + 1/4*(5*z^132 -
z^96 + z^72 - 5*z^48 - 5*z^36 + z^12 + 2)*x_3^2 + 1/4*(z^132 - 2*z^96 +
2*z^72 - z^48 - z^36 + 2*z^12)*x_3*x_4 + 1/4*(-5*z^132 + z^96 - z^72 +
5*z^48 + 5*z^36 - z^12 - 2)*x_3*x_5 + 1/4*(-3*z^132 - 5*z^96 + 5*z^72 +
3*z^48 + 3*z^36 + 5*z^12 + 4)*x_3*x_6 + 1/4*(-z^96 + z^72 + z^12 -
2)*x_4^2 + 1/4*(4*z^132 - 4*z^48 - 4*z^36 + 3)*x_4*x_5 + 1/4*(3*z^132 -
6*z^96 + 6*z^72 - 3*z^48 - 3*z^36 + 6*z^12 + 7)*x_4*x_6 + 1/4*(-z^132 +
z^96 - z^72 + z^48 + z^36 - z^12 + 1)*x_5^2 + 1/4*(z^132 - z^96 + z^72 -
z^48 - z^36 + z^12)*x_5*x_6 + 1/4*(3*z^132 - 3*z^96 + 3*z^72 - 3*z^48 -
3*z^36 + 3*z^12 + 7)*x_6^2,
x_1^2 + (-6*z^132 + 14*z^96 - 14*z^72 + 6*z^48 + 6*z^36 - 14*z^12 -
2)*x_1*x_3 + (-18*z^132 + 34*z^96 - 34*z^72 + 18*z^48 + 18*z^36 -
34*z^12 + 6)*x_1*x_4 + (18*z^132 - 30*z^96 + 30*z^72 - 18*z^48 - 18*z^36
+ 30*z^12 - 8)*x_1*x_5 + (-24*z^132 + 44*z^96 - 44*z^72 + 24*z^48 +
24*z^36 - 44*z^12 + 10)*x_1*x_6 + (7*z^132 - 12*z^96 + 12*z^72 - 7*z^48
- 7*z^36 + 12*z^12 - 2)*x_2^2 + (-2*z^132 + 2*z^96 - 2*z^72 + 2*z^48 +
2*z^36 - 2*z^12 + 4)*x_2*x_3 + (-2*z^132 + 2*z^96 - 2*z^72 + 2*z^48 +
2*z^36 - 2*z^12)*x_2*x_4 + (2*z^132 - 2*z^96 + 2*z^72 - 2*z^48 - 2*z^36
+ 2*z^12)*x_2*x_5 + (24*z^132 - 44*z^96 + 44*z^72 - 24*z^48 - 24*z^36 +
44*z^12 - 12)*x_2*x_6 + (10*z^132 - 17*z^96 + 17*z^72 - 10*z^48 -
10*z^36 + 17*z^12 - 7)*x_3^2 + (8*z^132 - 14*z^96 + 14*z^72 - 8*z^48 -
8*z^36 + 14*z^12 - 2)*x_3*x_4 + (-22*z^132 + 40*z^96 - 40*z^72 + 22*z^48
+ 22*z^36 - 40*z^12 + 12)*x_3*x_5 + (2*z^132 - 10*z^96 + 10*z^72 -
2*z^48 - 2*z^36 + 10*z^12 + 2)*x_3*x_6 + (10*z^132 - 17*z^96 + 17*z^72 -
10*z^48 - 10*z^36 + 17*z^12 - 3)*x_4^2 + (-22*z^132 + 40*z^96 - 40*z^72
+ 22*z^48 + 22*z^36 - 40*z^12 + 8)*x_4*x_5 + (14*z^132 - 30*z^96 +
30*z^72 - 14*z^48 - 14*z^36 + 30*z^12 - 6)*x_4*x_6 + (-8*z^132 + 13*z^96
- 13*z^72 + 8*z^48 + 8*z^36 - 13*z^12 + 4)*x_5^2 + (10*z^132 - 18*z^96 +
18*z^72 - 10*z^48 - 10*z^36 + 18*z^12 - 4)*x_5*x_6 + (-4*z^132 + 4*z^96
- 4*z^72 + 4*z^48 + 4*z^36 - 4*z^12 + 1)*x_6^2,
x_1*x_2 + (-3*z^132 + 6*z^96 - 6*z^72 + 3*z^48 + 3*z^36 - 6*z^12 -
1)*x_1*x_3 + (-7*z^132 + 14*z^96 - 14*z^72 + 7*z^48 + 7*z^36 - 14*z^12 +
3)*x_1*x_4 + (7*z^132 - 12*z^96 + 12*z^72 - 7*z^48 - 7*z^36 + 12*z^12 -
3)*x_1*x_5 + (-10*z^132 + 18*z^96 - 18*z^72 + 10*z^48 + 10*z^36 -
18*z^12 + 4)*x_1*x_6 + (3*z^132 - 5*z^96 + 5*z^72 - 3*z^48 - 3*z^36 +
5*z^12 - 1)*x_2^2 + (-z^132 + z^96 - z^72 + z^48 + z^36 - z^12 +
1)*x_2*x_3 + (-z^132 + z^96 - z^72 + z^48 + z^36 - z^12 - 1)*x_2*x_4 +
(z^132 - z^96 + z^72 - z^48 - z^36 + z^12)*x_2*x_5 + (10*z^132 - 18*z^96
+ 18*z^72 - 10*z^48 - 10*z^36 + 18*z^12 - 5)*x_2*x_6 + (4*z^132 - 7*z^96
+ 7*z^72 - 4*z^48 - 4*z^36 + 7*z^12 - 3)*x_3^2 + (4*z^132 - 6*z^96 +
6*z^72 - 4*z^48 - 4*z^36 + 6*z^12)*x_3*x_4 + (-9*z^132 + 16*z^96 -
16*z^72 + 9*z^48 + 9*z^36 - 16*z^12 + 5)*x_3*x_5 + (z^132 - 4*z^96 +
4*z^72 - z^48 - z^36 + 4*z^12 + 1)*x_3*x_6 + (4*z^132 - 7*z^96 + 7*z^72
- 4*z^48 - 4*z^36 + 7*z^12 - 1)*x_4^2 + (-9*z^132 + 16*z^96 - 16*z^72 +
9*z^48 + 9*z^36 - 16*z^12 + 3)*x_4*x_5 + (5*z^132 - 12*z^96 + 12*z^72 -
5*z^48 - 5*z^36 + 12*z^12 - 3)*x_4*x_6 + (-3*z^132 + 5*z^96 - 5*z^72 +
3*z^48 + 3*z^36 - 5*z^12 + 2)*x_5^2 + (5*z^132 - 8*z^96 + 8*z^72 -
5*z^48 - 5*z^36 + 8*z^12 - 1)*x_5*x_6 + (-2*z^132 + 2*z^96 - 2*z^72 +
2*z^48 + 2*z^36 - 2*z^12)*x_6^2
]
The output above shows that the ideal contains quadrics from two isotypical subspaces of \(S_2\). Note that the power of z\(=\zeta_{504}\) in our equations is always a multiple of 12. Therefore in the sequel we reduce these to z_42\(= \zeta_{42}\).
In both cases, the isotypical subspace of quadrics in the ideal is equal to the full isotypical subspace of quadrics in the polynomial ring. Thus, the polynomials shown are the candidate polynomials, and there are no unknown coefficients.
> K<z_42>:=CyclotomicField(42);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> x_0^2 + x_0*x_2 + x_1^2 - x_1*x_2 - x_1*x_4 - x_1*x_5 - x_1*x_6 + x_2^2 + x_\
2*x_5 + x_3^2 + x_3*x_4 + x_4^2 + x_4*x_6 + x_5^2 + x_6^2,
> x_0^2 + 1/2*(3*z_42^11 - 3*z_42^4 - 3*z_42^3 - 2)*x_1*x_3 + 1/2*(z_42^11 - z\
_42^8 + z_42^6 - z_42^4 - z_42^3 + z_42 - 1)*x_1*x_4 + 1/2*(2*z_42^11 + z_42^8\
- z_42^6 - 2*z_42^4 - 2*z_42^3 - z_42 + 1)*x_1*x_5 + 1/2*(3*z_42^11 - z_42^8 \
+ z_42^6 - 3*z_42^4 - 3*z_42^3 + z_42 + 1)*x_1*x_6 + 1/2*(-z_42^11 - 2*z_42^8 \
+ 2*z_42^6 + z_42^4 + z_42^3 + 2*z_42 + 3)*x_2*x_3 + 1/2*(-z_42^11 + z_42^4 + \
z_42^3)*x_2*x_4 - 1/2*x_2*x_5 + 1/2*(-3*z_42^11 + 2*z_42^8 - 2*z_42^6 + 3*z_42\
^4 + 3*z_42^3 - 2*z_42 - 2)*x_2*x_6 + 1/2*(z_42^11 + z_42^8 - z_42^6 - z_42^4 \
- z_42^3 - z_42 - 2)*x_3^2 + 1/2*(-z_42^11 + 2*z_42^8 - 2*z_42^6 + z_42^4 + z_\
42^3 - 2*z_42 - 2)*x_3*x_4 + 1/2*(-z_42^11 - z_42^8 + z_42^6 + z_42^4 + z_42^3\
+ z_42)*x_3*x_5 + 1/2*(-3*z_42^11 + z_42^8 - z_42^6 + 3*z_42^4 + 3*z_42^3 - z\
_42 + 2)*x_3*x_6 + 1/2*(z_42^8 - z_42^6 - z_42)*x_4^2 + 1/2*(-2*z_42^11 + 2*z_\
42^8 - 2*z_42^6 + 2*z_42^4 + 2*z_42^3 - 2*z_42 - 3)*x_4*x_5 + 1/2*(-z_42^11 + \
z_42^4 + z_42^3 + 3)*x_4*x_6 + 1/2*(-z_42^11 - z_42^8 + z_42^6 + z_42^4 + z_42\
^3 + z_42 - 1)*x_5^2 + 1/2*(-3*z_42^11 + z_42^8 - z_42^6 + 3*z_42^4 + 3*z_42^3\
- z_42 - 2)*x_5*x_6 + 1/2*(-z_42^11 - z_42^8 + z_42^6 + z_42^4 + z_42^3 + z_4\
2 + 1)*x_6^2,
> x_0*x_1 + 1/2*(-5*z_42^11 + 5*z_42^8 - 5*z_42^6 + 5*z_42^4 + 5*z_42^3 - 5*z_\
42 + 10)*x_1*x_3 + 1/2*(-16*z_42^11 + 27*z_42^8 - 27*z_42^6 + 16*z_42^4 + 16*z\
_42^3 - 27*z_42 + 8)*x_1*x_4 + 1/2*(11*z_42^11 - 26*z_42^8 + 26*z_42^6 - 11*z_\
42^4 - 11*z_42^3 + 26*z_42 - 8)*x_1*x_5 + 1/2*(-22*z_42^11 + 37*z_42^8 - 37*z_\
42^6 + 22*z_42^4 + 22*z_42^3 - 37*z_42 + 8)*x_1*x_6 + (2*z_42^11 - 4*z_42^8 + \
4*z_42^6 - 2*z_42^4 - 2*z_42^3 + 4*z_42 - 2)*x_2^2 + 1/2*(-2*z_42^11 + 7*z_42^\
8 - 7*z_42^6 + 2*z_42^4 + 2*z_42^3 - 7*z_42 - 5)*x_2*x_3 + 1/2*(z_42^11 + z_42\
^8 - z_42^6 - z_42^4 - z_42^3 - z_42 + 2)*x_2*x_4 + 1/2*(-z_42^11 + z_42^4 + z\
_42^3 - 1)*x_2*x_5 + 1/2*(23*z_42^11 - 39*z_42^8 + 39*z_42^6 - 23*z_42^4 - 23*\
z_42^3 + 39*z_42 - 6)*x_2*x_6 + 1/2*(9*z_42^11 - 17*z_42^8 + 17*z_42^6 - 9*z_4\
2^4 - 9*z_42^3 + 17*z_42 + 1)*x_3^2 + 1/2*(5*z_42^11 - 11*z_42^8 + 11*z_42^6 -\
5*z_42^4 - 5*z_42^3 + 11*z_42 - 4)*x_3*x_4 + 1/2*(-19*z_42^11 + 35*z_42^8 - 3\
5*z_42^6 + 19*z_42^4 + 19*z_42^3 - 35*z_42 + 5)*x_3*x_5 + 1/2*(5*z_42^11 - 3*z\
_42^8 + 3*z_42^6 - 5*z_42^4 - 5*z_42^3 + 3*z_42 - 7)*x_3*x_6 + 1/2*(6*z_42^11 \
- 14*z_42^8 + 14*z_42^6 - 6*z_42^4 - 6*z_42^3 + 14*z_42 - 5)*x_4^2 + 1/2*(-15*\
z_42^11 + 30*z_42^8 - 30*z_42^6 + 15*z_42^4 + 15*z_42^3 - 30*z_42 + 13)*x_4*x_\
5 + 1/2*(16*z_42^11 - 23*z_42^8 + 23*z_42^6 - 16*z_42^4 - 16*z_42^3 + 23*z_42 \
- 7)*x_4*x_6 + 1/2*(-6*z_42^11 + 13*z_42^8 - 13*z_42^6 + 6*z_42^4 + 6*z_42^3 -\
13*z_42 + 2)*x_5^2 + 1/2*(9*z_42^11 - 15*z_42^8 + 15*z_42^6 - 9*z_42^4 - 9*z_\
42^3 + 15*z_42 - 3)*x_5*x_6 + 1/2*(2*z_42^11 + 3*z_42^8 - 3*z_42^6 - 2*z_42^4 \
- 2*z_42^3 - 3*z_42 + 2)*x_6^2,
> x_0*x_2 + 1/2*(-9*z_42^11 + 8*z_42^8 - 8*z_42^6 + 9*z_42^4 + 9*z_42^3 - 8*z_\
42 + 6)*x_1*x_3 + 1/2*(-17*z_42^11 + 29*z_42^8 - 29*z_42^6 + 17*z_42^4 + 17*z_\
42^3 - 29*z_42 + 9)*x_1*x_4 + 1/2*(12*z_42^11 - 27*z_42^8 + 27*z_42^6 - 12*z_4\
2^4 - 12*z_42^3 + 27*z_42 - 7)*x_1*x_5 + 1/2*(-23*z_42^11 + 39*z_42^8 - 39*z_4\
2^6 + 23*z_42^4 + 23*z_42^3 - 39*z_42 + 9)*x_1*x_6 + (3*z_42^11 - 5*z_42^8 + 5\
*z_42^6 - 3*z_42^4 - 3*z_42^3 + 5*z_42 - 1)*x_2^2 + 1/2*(-z_42^11 + 6*z_42^8 -\
6*z_42^6 + z_42^4 + z_42^3 - 6*z_42 - 3)*x_2*x_3 + 1/2*(-z_42^11 + 2*z_42^8 -\
2*z_42^6 + z_42^4 + z_42^3 - 2*z_42)*x_2*x_4 + 1/2*(2*z_42^11 - 2*z_42^8 + 2*\
z_42^6 - 2*z_42^4 - 2*z_42^3 + 2*z_42 + 1)*x_2*x_5 + 1/2*(23*z_42^11 - 40*z_42\
^8 + 40*z_42^6 - 23*z_42^4 - 23*z_42^3 + 40*z_42 - 8)*x_2*x_6 + 1/2*(7*z_42^11\
- 17*z_42^8 + 17*z_42^6 - 7*z_42^4 - 7*z_42^3 + 17*z_42 - 2)*x_3^2 + 1/2*(7*z\
_42^11 - 12*z_42^8 + 12*z_42^6 - 7*z_42^4 - 7*z_42^3 + 12*z_42 - 2)*x_3*x_4 + \
1/2*(-19*z_42^11 + 37*z_42^8 - 37*z_42^6 + 19*z_42^4 + 19*z_42^3 - 37*z_42 + 8\
)*x_3*x_5 + 1/2*(7*z_42^11 - 5*z_42^8 + 5*z_42^6 - 7*z_42^4 - 7*z_42^3 + 5*z_4\
2 - 6)*x_3*x_6 + 1/2*(8*z_42^11 - 15*z_42^8 + 15*z_42^6 - 8*z_42^4 - 8*z_42^3 \
+ 15*z_42 - 4)*x_4^2 + 1/2*(-18*z_42^11 + 32*z_42^8 - 32*z_42^6 + 18*z_42^4 + \
18*z_42^3 - 32*z_42 + 11)*x_4*x_5 + 1/2*(15*z_42^11 - 24*z_42^8 + 24*z_42^6 - \
15*z_42^4 - 15*z_42^3 + 24*z_42 - 11)*x_4*x_6 + 1/2*(-5*z_42^11 + 13*z_42^8 - \
13*z_42^6 + 5*z_42^4 + 5*z_42^3 - 13*z_42 + 3)*x_5^2 + 1/2*(11*z_42^11 - 17*z_\
42^8 + 17*z_42^6 - 11*z_42^4 - 11*z_42^3 + 17*z_42 - 2)*x_5*x_6 + 1/2*(-z_42^1\
1 + 5*z_42^8 - 5*z_42^6 + z_42^4 + z_42^3 - 5*z_42 - 1)*x_6^2,
> x_0*x_3 + 1/4*(z_42^11 - 2*z_42^8 + 2*z_42^6 - z_42^4 - z_42^3 + 2*z_42 - 2)\
*x_1*x_3 + 1/4*(5*z_42^11 - 11*z_42^8 + 11*z_42^6 - 5*z_42^4 - 5*z_42^3 + 11*z\
_42 - 5)*x_1*x_4 + 1/4*(-4*z_42^11 + 13*z_42^8 - 13*z_42^6 + 4*z_42^4 + 4*z_42\
^3 - 13*z_42 + 5)*x_1*x_5 + 1/4*(11*z_42^11 - 19*z_42^8 + 19*z_42^6 - 11*z_42^\
4 - 11*z_42^3 + 19*z_42 - 3)*x_1*x_6 + 1/2*(-z_42^11 + 2*z_42^8 - 2*z_42^6 + z\
_42^4 + z_42^3 - 2*z_42 + 1)*x_2^2 + 1/4*(3*z_42^11 - 4*z_42^8 + 4*z_42^6 - 3*\
z_42^4 - 3*z_42^3 + 4*z_42 + 5)*x_2*x_3 + 1/4*(z_42^11 - 2*z_42^8 + 2*z_42^6 -\
z_42^4 - z_42^3 + 2*z_42)*x_2*x_4 + 1/4*(-2*z_42^11 + 2*z_42^4 + 2*z_42^3 + 1\
)*x_2*x_5 + 1/4*(-13*z_42^11 + 20*z_42^8 - 20*z_42^6 + 13*z_42^4 + 13*z_42^3 -\
20*z_42 + 2)*x_2*x_6 + 1/4*(-5*z_42^11 + 7*z_42^8 - 7*z_42^6 + 5*z_42^4 + 5*z\
_42^3 - 7*z_42 + 2)*x_3^2 + 1/4*(-3*z_42^11 + 4*z_42^8 - 4*z_42^6 + 3*z_42^4 +\
3*z_42^3 - 4*z_42 + 2)*x_3*x_4 + 1/4*(9*z_42^11 - 19*z_42^8 + 19*z_42^6 - 9*z\
_42^4 - 9*z_42^3 + 19*z_42 - 2)*x_3*x_5 + 1/4*(-z_42^11 + 3*z_42^8 - 3*z_42^6 \
+ z_42^4 + z_42^3 - 3*z_42)*x_3*x_6 + 1/4*(-2*z_42^11 + 5*z_42^8 - 5*z_42^6 + \
2*z_42^4 + 2*z_42^3 - 5*z_42 + 4)*x_4^2 + 1/4*(8*z_42^11 - 18*z_42^8 + 18*z_42\
^6 - 8*z_42^4 - 8*z_42^3 + 18*z_42 - 5)*x_4*x_5 + 1/4*(-5*z_42^11 + 10*z_42^8 \
- 10*z_42^6 + 5*z_42^4 + 5*z_42^3 - 10*z_42 + 5)*x_4*x_6 + 1/4*(z_42^11 - 5*z_\
42^8 + 5*z_42^6 - z_42^4 - z_42^3 + 5*z_42 - 3)*x_5^2 + 1/4*(-5*z_42^11 + 7*z_\
42^8 - 7*z_42^6 + 5*z_42^4 + 5*z_42^3 - 7*z_42)*x_5*x_6 + 1/4*(z_42^11 - z_42^\
8 + z_42^6 - z_42^4 - z_42^3 + z_42 - 1)*x_6^2,
> x_0*x_4 + 1/2*(-9*z_42^11 + 12*z_42^8 - 12*z_42^6 + 9*z_42^4 + 9*z_42^3 - 12\
*z_42 + 7)*x_1*x_3 + 1/2*(-23*z_42^11 + 42*z_42^8 - 42*z_42^6 + 23*z_42^4 + 23\
*z_42^3 - 42*z_42 + 12)*x_1*x_4 + 1/2*(20*z_42^11 - 40*z_42^8 + 40*z_42^6 - 20\
*z_42^4 - 20*z_42^3 + 40*z_42 - 9)*x_1*x_5 + 1/2*(-31*z_42^11 + 56*z_42^8 - 56\
*z_42^6 + 31*z_42^4 + 31*z_42^3 - 56*z_42 + 14)*x_1*x_6 + (4*z_42^11 - 7*z_42^\
8 + 7*z_42^6 - 4*z_42^4 - 4*z_42^3 + 7*z_42 - 2)*x_2^2 + 1/2*(-2*z_42^11 + 7*z\
_42^8 - 7*z_42^6 + 2*z_42^4 + 2*z_42^3 - 7*z_42 - 2)*x_2*x_3 + 1/2*(-z_42^11 +\
2*z_42^8 - 2*z_42^6 + z_42^4 + z_42^3 - 2*z_42 + 1)*x_2*x_4 + 1/2*(z_42^11 - \
z_42^8 + z_42^6 - z_42^4 - z_42^3 + z_42 - 1)*x_2*x_5 + 1/2*(33*z_42^11 - 58*z\
_42^8 + 58*z_42^6 - 33*z_42^4 - 33*z_42^3 + 58*z_42 - 13)*x_2*x_6 + 1/2*(12*z_\
42^11 - 23*z_42^8 + 23*z_42^6 - 12*z_42^4 - 12*z_42^3 + 23*z_42 - 5)*x_3^2 + 1\
/2*(9*z_42^11 - 16*z_42^8 + 16*z_42^6 - 9*z_42^4 - 9*z_42^3 + 16*z_42 - 5)*x_3\
*x_4 + 1/2*(-28*z_42^11 + 53*z_42^8 - 53*z_42^6 + 28*z_42^4 + 28*z_42^3 - 53*z\
_42 + 11)*x_3*x_5 + 1/2*(6*z_42^11 - 7*z_42^8 + 7*z_42^6 - 6*z_42^4 - 6*z_42^3\
+ 7*z_42 - 5)*x_3*x_6 + 1/2*(11*z_42^11 - 21*z_42^8 + 21*z_42^6 - 11*z_42^4 -\
11*z_42^3 + 21*z_42 - 6)*x_4^2 + 1/2*(-27*z_42^11 + 49*z_42^8 - 49*z_42^6 + 2\
7*z_42^4 + 27*z_42^3 - 49*z_42 + 13)*x_4*x_5 + 1/2*(20*z_42^11 - 35*z_42^8 + 3\
5*z_42^6 - 20*z_42^4 - 20*z_42^3 + 35*z_42 - 12)*x_4*x_6 + 1/2*(-9*z_42^11 + 1\
8*z_42^8 - 18*z_42^6 + 9*z_42^4 + 9*z_42^3 - 18*z_42 + 4)*x_5^2 + 1/2*(14*z_42\
^11 - 23*z_42^8 + 23*z_42^6 - 14*z_42^4 - 14*z_42^3 + 23*z_42 - 5)*x_5*x_6 + 1\
/2*(-3*z_42^11 + 6*z_42^8 - 6*z_42^6 + 3*z_42^4 + 3*z_42^3 - 6*z_42)*x_6^2,
> x_0*x_5 + 1/4*(-11*z_42^11 + 22*z_42^8 - 22*z_42^6 + 11*z_42^4 + 11*z_42^3 -\
22*z_42 + 4)*x_1*x_3 + 1/4*(-35*z_42^11 + 63*z_42^8 - 63*z_42^6 + 35*z_42^4 +\
35*z_42^3 - 63*z_42 + 13)*x_1*x_4 + 1/4*(32*z_42^11 - 57*z_42^8 + 57*z_42^6 -\
32*z_42^4 - 32*z_42^3 + 57*z_42 - 15)*x_1*x_5 + 1/4*(-45*z_42^11 + 83*z_42^8 \
- 83*z_42^6 + 45*z_42^4 + 45*z_42^3 - 83*z_42 + 19)*x_1*x_6 + 1/2*(5*z_42^11 -\
10*z_42^8 + 10*z_42^6 - 5*z_42^4 - 5*z_42^3 + 10*z_42 - 3)*x_2^2 + 1/4*(-3*z_\
42^11 + 6*z_42^8 - 6*z_42^6 + 3*z_42^4 + 3*z_42^3 - 6*z_42 + 3)*x_2*x_3 + 1/4*\
(z_42^11 + 2*z_42^8 - 2*z_42^6 - z_42^4 - z_42^3 - 2*z_42 + 2)*x_2*x_4 + 1/4*(\
-2*z_42^8 + 2*z_42^6 + 2*z_42 + 1)*x_2*x_5 + 1/4*(47*z_42^11 - 84*z_42^8 + 84*\
z_42^6 - 47*z_42^4 - 47*z_42^3 + 84*z_42 - 20)*x_2*x_6 + 1/4*(21*z_42^11 - 33*\
z_42^8 + 33*z_42^6 - 21*z_42^4 - 21*z_42^3 + 33*z_42 - 8)*x_3^2 + 1/4*(13*z_42\
^11 - 24*z_42^8 + 24*z_42^6 - 13*z_42^4 - 13*z_42^3 + 24*z_42 - 8)*x_3*x_4 + 1\
/4*(-41*z_42^11 + 73*z_42^8 - 73*z_42^6 + 41*z_42^4 + 41*z_42^3 - 73*z_42 + 20\
)*x_3*x_5 + 1/4*(5*z_42^11 - 13*z_42^8 + 13*z_42^6 - 5*z_42^4 - 5*z_42^3 + 13*\
z_42 - 2)*x_3*x_6 + 1/4*(16*z_42^11 - 31*z_42^8 + 31*z_42^6 - 16*z_42^4 - 16*z\
_42^3 + 31*z_42 - 8)*x_4^2 + 1/4*(-38*z_42^11 + 72*z_42^8 - 72*z_42^6 + 38*z_4\
2^4 + 38*z_42^3 - 72*z_42 + 19)*x_4*x_5 + 1/4*(29*z_42^11 - 56*z_42^8 + 56*z_4\
2^6 - 29*z_42^4 - 29*z_42^3 + 56*z_42 - 9)*x_4*x_6 + 1/4*(-15*z_42^11 + 25*z_4\
2^8 - 25*z_42^6 + 15*z_42^4 + 15*z_42^3 - 25*z_42 + 7)*x_5^2 + 1/4*(17*z_42^11\
- 33*z_42^8 + 33*z_42^6 - 17*z_42^4 - 17*z_42^3 + 33*z_42 - 10)*x_5*x_6 + 1/4\
*(-3*z_42^11 + 5*z_42^8 - 5*z_42^6 + 3*z_42^4 + 3*z_42^3 - 5*z_42 + 5)*x_6^2,
> x_0*x_6 + 1/4*(5*z_42^11 + 4*z_42^8 - 4*z_42^6 - 5*z_42^4 - 5*z_42^3 - 4*z_4\
2)*x_1*x_3 + 1/4*(-z_42^11 + 3*z_42^8 - 3*z_42^6 + z_42^4 + z_42^3 - 3*z_42 - \
1)*x_1*x_4 + 1/4*(-2*z_42^11 + z_42^8 - z_42^6 + 2*z_42^4 + 2*z_42^3 - z_42 - \
5)*x_1*x_5 + 1/4*(-3*z_42^11 + 3*z_42^8 - 3*z_42^6 + 3*z_42^4 + 3*z_42^3 - 3*z\
_42 - 3)*x_1*x_6 + 1/2*(-z_42^11 + z_42^4 + z_42^3 - 1)*x_2^2 + 1/4*(-5*z_42^1\
1 + 5*z_42^4 + 5*z_42^3 - 1)*x_2*x_3 + 1/4*(z_42^11 - z_42^4 - z_42^3 + 2)*x_2\
*x_4 + 1/4*(-2*z_42^11 + 2*z_42^8 - 2*z_42^6 + 2*z_42^4 + 2*z_42^3 - 2*z_42 - \
3)*x_2*x_5 + 1/4*(3*z_42^11 - 2*z_42^8 + 2*z_42^6 - 3*z_42^4 - 3*z_42^3 + 2*z_\
42 + 4)*x_2*x_6 + 1/4*(5*z_42^11 - z_42^8 + z_42^6 - 5*z_42^4 - 5*z_42^3 + z_4\
2 + 2)*x_3^2 + 1/4*(z_42^11 - 2*z_42^8 + 2*z_42^6 - z_42^4 - z_42^3 + 2*z_42)*\
x_3*x_4 + 1/4*(-5*z_42^11 + z_42^8 - z_42^6 + 5*z_42^4 + 5*z_42^3 - z_42 - 2)*\
x_3*x_5 + 1/4*(-3*z_42^11 - 5*z_42^8 + 5*z_42^6 + 3*z_42^4 + 3*z_42^3 + 5*z_42\
+ 4)*x_3*x_6 + 1/4*(-z_42^8 + z_42^6 + z_42 - 2)*x_4^2 + 1/4*(4*z_42^11 - 4*z\
_42^4 - 4*z_42^3 + 3)*x_4*x_5 + 1/4*(3*z_42^11 - 6*z_42^8 + 6*z_42^6 - 3*z_42^\
4 - 3*z_42^3 + 6*z_42 + 7)*x_4*x_6 + 1/4*(-z_42^11 + z_42^8 - z_42^6 + z_42^4 \
+ z_42^3 - z_42 + 1)*x_5^2 + 1/4*(z_42^11 - z_42^8 + z_42^6 - z_42^4 - z_42^3 \
+ z_42)*x_5*x_6 + 1/4*(3*z_42^11 - 3*z_42^8 + 3*z_42^6 - 3*z_42^4 - 3*z_42^3 +\
3*z_42 + 7)*x_6^2,
> x_1^2 + (-6*z_42^11 + 14*z_42^8 - 14*z_42^6 + 6*z_42^4 + 6*z_42^3 - 14*z_42 \
- 2)*x_1*x_3 + (-18*z_42^11 + 34*z_42^8 - 34*z_42^6 + 18*z_42^4 + 18*z_42^3 - \
34*z_42 + 6)*x_1*x_4 + (18*z_42^11 - 30*z_42^8 + 30*z_42^6 - 18*z_42^4 - 18*z_\
42^3 + 30*z_42 - 8)*x_1*x_5 + (-24*z_42^11 + 44*z_42^8 - 44*z_42^6 + 24*z_42^4\
+ 24*z_42^3 - 44*z_42 + 10)*x_1*x_6 + (7*z_42^11 - 12*z_42^8 + 12*z_42^6 - 7*\
z_42^4 - 7*z_42^3 + 12*z_42 - 2)*x_2^2 + (-2*z_42^11 + 2*z_42^8 - 2*z_42^6 + 2\
*z_42^4 + 2*z_42^3 - 2*z_42 + 4)*x_2*x_3 + (-2*z_42^11 + 2*z_42^8 - 2*z_42^6 +\
2*z_42^4 + 2*z_42^3 - 2*z_42)*x_2*x_4 + (2*z_42^11 - 2*z_42^8 + 2*z_42^6 - 2*\
z_42^4 - 2*z_42^3 + 2*z_42)*x_2*x_5 + (24*z_42^11 - 44*z_42^8 + 44*z_42^6 - 24\
*z_42^4 - 24*z_42^3 + 44*z_42 - 12)*x_2*x_6 + (10*z_42^11 - 17*z_42^8 + 17*z_4\
2^6 - 10*z_42^4 - 10*z_42^3 + 17*z_42 - 7)*x_3^2 + (8*z_42^11 - 14*z_42^8 + 14\
*z_42^6 - 8*z_42^4 - 8*z_42^3 + 14*z_42 - 2)*x_3*x_4 + (-22*z_42^11 + 40*z_42^\
8 - 40*z_42^6 + 22*z_42^4 + 22*z_42^3 - 40*z_42 + 12)*x_3*x_5 + (2*z_42^11 - 1\
0*z_42^8 + 10*z_42^6 - 2*z_42^4 - 2*z_42^3 + 10*z_42 + 2)*x_3*x_6 + (10*z_42^1\
1 - 17*z_42^8 + 17*z_42^6 - 10*z_42^4 - 10*z_42^3 + 17*z_42 - 3)*x_4^2 + (-22*\
z_42^11 + 40*z_42^8 - 40*z_42^6 + 22*z_42^4 + 22*z_42^3 - 40*z_42 + 8)*x_4*x_5\
+ (14*z_42^11 - 30*z_42^8 + 30*z_42^6 - 14*z_42^4 - 14*z_42^3 + 30*z_42 - 6)*\
x_4*x_6 + (-8*z_42^11 + 13*z_42^8 - 13*z_42^6 + 8*z_42^4 + 8*z_42^3 - 13*z_42 \
+ 4)*x_5^2 + (10*z_42^11 - 18*z_42^8 + 18*z_42^6 - 10*z_42^4 - 10*z_42^3 + 18*\
z_42 - 4)*x_5*x_6 + (-4*z_42^11 + 4*z_42^8 - 4*z_42^6 + 4*z_42^4 + 4*z_42^3 - \
4*z_42 + 1)*x_6^2,
> x_1*x_2 + (-3*z_42^11 + 6*z_42^8 - 6*z_42^6 + 3*z_42^4 + 3*z_42^3 - 6*z_42 -\
1)*x_1*x_3 + (-7*z_42^11 + 14*z_42^8 - 14*z_42^6 + 7*z_42^4 + 7*z_42^3 - 14*z\
_42 + 3)*x_1*x_4 + (7*z_42^11 - 12*z_42^8 + 12*z_42^6 - 7*z_42^4 - 7*z_42^3 + \
12*z_42 - 3)*x_1*x_5 + (-10*z_42^11 + 18*z_42^8 - 18*z_42^6 + 10*z_42^4 + 10*z\
_42^3 - 18*z_42 + 4)*x_1*x_6 + (3*z_42^11 - 5*z_42^8 + 5*z_42^6 - 3*z_42^4 - 3\
*z_42^3 + 5*z_42 - 1)*x_2^2 + (-z_42^11 + z_42^8 - z_42^6 + z_42^4 + z_42^3 - \
z_42 + 1)*x_2*x_3 + (-z_42^11 + z_42^8 - z_42^6 + z_42^4 + z_42^3 - z_42 - 1)*\
x_2*x_4 + (z_42^11 - z_42^8 + z_42^6 - z_42^4 - z_42^3 + z_42)*x_2*x_5 + (10*z\
_42^11 - 18*z_42^8 + 18*z_42^6 - 10*z_42^4 - 10*z_42^3 + 18*z_42 - 5)*x_2*x_6 \
+ (4*z_42^11 - 7*z_42^8 + 7*z_42^6 - 4*z_42^4 - 4*z_42^3 + 7*z_42 - 3)*x_3^2 +\
(4*z_42^11 - 6*z_42^8 + 6*z_42^6 - 4*z_42^4 - 4*z_42^3 + 6*z_42)*x_3*x_4 + (-\
9*z_42^11 + 16*z_42^8 - 16*z_42^6 + 9*z_42^4 + 9*z_42^3 - 16*z_42 + 5)*x_3*x_5\
+ (z_42^11 - 4*z_42^8 + 4*z_42^6 - z_42^4 - z_42^3 + 4*z_42 + 1)*x_3*x_6 + (4\
*z_42^11 - 7*z_42^8 + 7*z_42^6 - 4*z_42^4 - 4*z_42^3 + 7*z_42 - 1)*x_4^2 + (-9\
*z_42^11 + 16*z_42^8 - 16*z_42^6 + 9*z_42^4 + 9*z_42^3 - 16*z_42 + 3)*x_4*x_5 \
+ (5*z_42^11 - 12*z_42^8 + 12*z_42^6 - 5*z_42^4 - 5*z_42^3 + 12*z_42 - 3)*x_4*\
x_6 + (-3*z_42^11 + 5*z_42^8 - 5*z_42^6 + 3*z_42^4 + 3*z_42^3 - 5*z_42 + 2)*x_\
5^2 + (5*z_42^11 - 8*z_42^8 + 8*z_42^6 - 5*z_42^4 - 5*z_42^3 + 8*z_42 - 1)*x_5\
*x_6 + (-2*z_42^11 + 2*z_42^8 - 2*z_42^6 + 2*z_42^4 + 2*z_42^3 - 2*z_42)*x_6^2
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> A:=Matrix([
> [-1/2,1,0,-1/2,1,1/2,-1/2],
> [1/2,1,0,-1/2,0,1/2,1/2],
> [-1/2,0,0,-1/2,1,-1/2,-1/2],
> [1/2,-1,-1,-1/2,0,1/2,-1/2],
> [0,-1,0,0,0,0,-1],
> [1/2,-1,-1,1/2,0,-1/2,-1/2],
> [-1/2,-1,0,1/2,-1,-1/2,-1/2]
> ]);
> B:=Matrix([
> [-1/2,0,0,-1/2,0,-1/2,1/2],
> [1/2,-1,-1,1/2,0,-1/2,-1/2],
> [0,0,0,0,-1,0,1],
> [1/2,-1,0,-1/2,0,-1/2,-1/2],
> [-1/2,0,1,-1/2,0,-1/2,-1/2],
> [0,1,1,0,0,0,1],
> [-1,0,1,0,0,0,0]
> ]);
> Order(A);
2
> Order(B);
3
> Order( (A*B)^-1);
7
> GL7K:=GeneralLinearGroup(7,K);
> IdentifyGroup(sub<GL7K | A,B>);
<504, 156>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations :
-1/2*x_0 + 1/2*x_1 - 1/2*x_2 + 1/2*x_3 + 1/2*x_5 - 1/2*x_6
x_0 + x_1 - x_3 - x_4 - x_5 - x_6
-x_3 - x_5
-1/2*x_0 - 1/2*x_1 - 1/2*x_2 - 1/2*x_3 + 1/2*x_5 + 1/2*x_6
x_0 + x_2 - x_6
1/2*x_0 + 1/2*x_1 - 1/2*x_2 + 1/2*x_3 - 1/2*x_5 - 1/2*x_6
-1/2*x_0 + 1/2*x_1 - 1/2*x_2 - 1/2*x_3 - x_4 - 1/2*x_5 - 1/2*x_6
and inverse
-1/2*x_0 + 1/2*x_1 - 1/2*x_2 + 1/2*x_3 + 1/2*x_5 - 1/2*x_6
x_0 + x_1 - x_3 - x_4 - x_5 - x_6
-x_3 - x_5
-1/2*x_0 - 1/2*x_1 - 1/2*x_2 - 1/2*x_3 + 1/2*x_5 + 1/2*x_6
x_0 + x_2 - x_6
1/2*x_0 + 1/2*x_1 - 1/2*x_2 + 1/2*x_3 - 1/2*x_5 - 1/2*x_6
-1/2*x_0 + 1/2*x_1 - 1/2*x_2 - 1/2*x_3 - x_4 - 1/2*x_5 - 1/2*x_6
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations :
-1/2*x_0 + 1/2*x_1 + 1/2*x_3 - 1/2*x_4 - x_6
-x_1 - x_3 + x_5
-x_1 + x_4 + x_5 + x_6
-1/2*x_0 + 1/2*x_1 - 1/2*x_3 - 1/2*x_4
-x_2
-1/2*x_0 - 1/2*x_1 - 1/2*x_3 - 1/2*x_4
1/2*x_0 - 1/2*x_1 + x_2 - 1/2*x_3 - 1/2*x_4 + x_5
and inverse
-1/2*x_0 - 1/2*x_2 - 1/2*x_3 + 1/2*x_4 - 1/2*x_5 + 1/2*x_6
x_3 - x_5
-x_4
1/2*x_0 - x_1 + 1/2*x_2 - 1/2*x_3 + 1/2*x_4 + 1/2*x_5 + 1/2*x_6
x_1 - x_4 - x_5 - x_6
1/2*x_0 + 1/2*x_2 + 1/2*x_3 + 1/2*x_4 - 1/2*x_5 + 1/2*x_6
-1/2*x_0 - x_1 + 1/2*x_2 + 1/2*x_3 + 1/2*x_4 + 1/2*x_5 + 1/2*x_6
Macaulay2, version 1.8.1
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
PrimaryDecomposition, ReesAlgebra, TangentCone
i1 : loadPackage("Cyclotomic");
i2 : K=cyclotomicField(42);
i3 : z_42=K_0;
i4 : T=K[x_0..x_6];
i5 : I=ideal({
x_0^2 + x_0*x_2 + x_1^2 - x_1*x_2 - x_1*x_4 - x_1*x_5 - x_1*x_6 + x_2^2 + x_2*x_5 + x_3^2 + x_3*x_4 + x_4^2 + x_4*x_6 + x_5^2 + x_6^2,
x_0^2 + 1/2*(3*z_42^11 - 3*z_42^4 - 3*z_42^3 - 2)*x_1*x_3 + 1/2*(z_42^11 - z_42^8 + z_42^6 - z_42^4 - z_42^3 + z_42 - 1)*x_1*x_4 + 1/2*(2*z_42^11 + z_42^8 - z_42^6 - 2*z_42^4 - 2*z_42^3 - z_42 + 1)*x_1*x_5 + 1/2*(3*z_42^11 - z_42^8 + z_42^6 - 3*z_42^4 - 3*z_42^3 + z_42 + 1)*x_1*x_6 + 1/2*(-z_42^11 - 2*z_42^8 + 2*z_42^6 + z_42^4 + z_42^3 + 2*z_42 + 3)*x_2*x_3 + 1/2*(-z_42^11 + z_42^4 + z_42^3)*x_2*x_4 - 1/2*x_2*x_5 + 1/2*(-3*z_42^11 + 2*z_42^8 - 2*z_42^6 + 3*z_42^4 + 3*z_42^3 - 2*z_42 - 2)*x_2*x_6 + 1/2*(z_42^11 + z_42^8 - z_42^6 - z_42^4 - z_42^3 - z_42 - 2)*x_3^2 + 1/2*(-z_42^11 + 2*z_42^8 - 2*z_42^6 + z_42^4 + z_42^3 - 2*z_42 - 2)*x_3*x_4 + 1/2*(-z_42^11 - z_42^8 + z_42^6 + z_42^4 + z_42^3 + z_42)*x_3*x_5 + 1/2*(-3*z_42^11 + z_42^8 - z_42^6 + 3*z_42^4 + 3*z_42^3 - z_42 + 2)*x_3*x_6 + 1/2*(z_42^8 - z_42^6 - z_42)*x_4^2 + 1/2*(-2*z_42^11 + 2*z_42^8 - 2*z_42^6 + 2*z_42^4 + 2*z_42^3 - 2*z_42 - 3)*x_4*x_5 + 1/2*(-z_42^11 + z_42^4 + z_42^3 + 3)*x_4*x_6 + 1/2*(-z_42^11 - z_42^8 + z_42^6 + z_42^4 + z_42^3 + z_42 - 1)*x_5^2 + 1/2*(-3*z_42^11 + z_42^8 - z_42^6 + 3*z_42^4 + 3*z_42^3 - z_42 - 2)*x_5*x_6 + 1/2*(-z_42^11 - z_42^8 + z_42^6 + z_42^4 + z_42^3 + z_42 + 1)*x_6^2,
x_0*x_1 + 1/2*(-5*z_42^11 + 5*z_42^8 - 5*z_42^6 + 5*z_42^4 + 5*z_42^3 - 5*z_42 + 10)*x_1*x_3 + 1/2*(-16*z_42^11 + 27*z_42^8 - 27*z_42^6 + 16*z_42^4 + 16*z_42^3 - 27*z_42 + 8)*x_1*x_4 + 1/2*(11*z_42^11 - 26*z_42^8 + 26*z_42^6 - 11*z_42^4 - 11*z_42^3 + 26*z_42 - 8)*x_1*x_5 + 1/2*(-22*z_42^11 + 37*z_42^8 - 37*z_42^6 + 22*z_42^4 + 22*z_42^3 - 37*z_42 + 8)*x_1*x_6 + (2*z_42^11 - 4*z_42^8 + 4*z_42^6 - 2*z_42^4 - 2*z_42^3 + 4*z_42 - 2)*x_2^2 + 1/2*(-2*z_42^11 + 7*z_42^8 - 7*z_42^6 + 2*z_42^4 + 2*z_42^3 - 7*z_42 - 5)*x_2*x_3 + 1/2*(z_42^11 + z_42^8 - z_42^6 - z_42^4 - z_42^3 - z_42 + 2)*x_2*x_4 + 1/2*(-z_42^11 + z_42^4 + z_42^3 - 1)*x_2*x_5 + 1/2*(23*z_42^11 - 39*z_42^8 + 39*z_42^6 - 23*z_42^4 - 23*z_42^3 + 39*z_42 - 6)*x_2*x_6 + 1/2*(9*z_42^11 - 17*z_42^8 + 17*z_42^6 - 9*z_42^4 - 9*z_42^3 + 17*z_42 + 1)*x_3^2 + 1/2*(5*z_42^11 - 11*z_42^8 + 11*z_42^6 - 5*z_42^4 - 5*z_42^3 + 11*z_42 - 4)*x_3*x_4 + 1/2*(-19*z_42^11 + 35*z_42^8 - 35*z_42^6 + 19*z_42^4 + 19*z_42^3 - 35*z_42 + 5)*x_3*x_5 + 1/2*(5*z_42^11 - 3*z_42^8 + 3*z_42^6 - 5*z_42^4 - 5*z_42^3 + 3*z_42 - 7)*x_3*x_6 + 1/2*(6*z_42^11 - 14*z_42^8 + 14*z_42^6 - 6*z_42^4 - 6*z_42^3 + 14*z_42 - 5)*x_4^2 + 1/2*(-15*z_42^11 + 30*z_42^8 - 30*z_42^6 + 15*z_42^4 + 15*z_42^3 - 30*z_42 + 13)*x_4*x_5 + 1/2*(16*z_42^11 - 23*z_42^8 + 23*z_42^6 - 16*z_42^4 - 16*z_42^3 + 23*z_42 - 7)*x_4*x_6 + 1/2*(-6*z_42^11 + 13*z_42^8 - 13*z_42^6 + 6*z_42^4 + 6*z_42^3 - 13*z_42 + 2)*x_5^2 + 1/2*(9*z_42^11 - 15*z_42^8 + 15*z_42^6 - 9*z_42^4 - 9*z_42^3 + 15*z_42 - 3)*x_5*x_6 + 1/2*(2*z_42^11 + 3*z_42^8 - 3*z_42^6 - 2*z_42^4 - 2*z_42^3 - 3*z_42 + 2)*x_6^2,
x_0*x_2 + 1/2*(-9*z_42^11 + 8*z_42^8 - 8*z_42^6 + 9*z_42^4 + 9*z_42^3 - 8*z_42 + 6)*x_1*x_3 + 1/2*(-17*z_42^11 + 29*z_42^8 - 29*z_42^6 + 17*z_42^4 + 17*z_42^3 - 29*z_42 + 9)*x_1*x_4 + 1/2*(12*z_42^11 - 27*z_42^8 + 27*z_42^6 - 12*z_42^4 - 12*z_42^3 + 27*z_42 - 7)*x_1*x_5 + 1/2*(-23*z_42^11 + 39*z_42^8 - 39*z_42^6 + 23*z_42^4 + 23*z_42^3 - 39*z_42 + 9)*x_1*x_6 + (3*z_42^11 - 5*z_42^8 + 5*z_42^6 - 3*z_42^4 - 3*z_42^3 + 5*z_42 - 1)*x_2^2 + 1/2*(-z_42^11 + 6*z_42^8 - 6*z_42^6 + z_42^4 + z_42^3 - 6*z_42 - 3)*x_2*x_3 + 1/2*(-z_42^11 + 2*z_42^8 - 2*z_42^6 + z_42^4 + z_42^3 - 2*z_42)*x_2*x_4 + 1/2*(2*z_42^11 - 2*z_42^8 + 2*z_42^6 - 2*z_42^4 - 2*z_42^3 + 2*z_42 + 1)*x_2*x_5 + 1/2*(23*z_42^11 - 40*z_42^8 + 40*z_42^6 - 23*z_42^4 - 23*z_42^3 + 40*z_42 - 8)*x_2*x_6 + 1/2*(7*z_42^11 - 17*z_42^8 + 17*z_42^6 - 7*z_42^4 - 7*z_42^3 + 17*z_42 - 2)*x_3^2 + 1/2*(7*z_42^11 - 12*z_42^8 + 12*z_42^6 - 7*z_42^4 - 7*z_42^3 + 12*z_42 - 2)*x_3*x_4 + 1/2*(-19*z_42^11 + 37*z_42^8 - 37*z_42^6 + 19*z_42^4 + 19*z_42^3 - 37*z_42 + 8)*x_3*x_5 + 1/2*(7*z_42^11 - 5*z_42^8 + 5*z_42^6 - 7*z_42^4 - 7*z_42^3 + 5*z_42 - 6)*x_3*x_6 + 1/2*(8*z_42^11 - 15*z_42^8 + 15*z_42^6 - 8*z_42^4 - 8*z_42^3 + 15*z_42 - 4)*x_4^2 + 1/2*(-18*z_42^11 + 32*z_42^8 - 32*z_42^6 + 18*z_42^4 + 18*z_42^3 - 32*z_42 + 11)*x_4*x_5 + 1/2*(15*z_42^11 - 24*z_42^8 + 24*z_42^6 - 15*z_42^4 - 15*z_42^3 + 24*z_42 - 11)*x_4*x_6 + 1/2*(-5*z_42^11 + 13*z_42^8 - 13*z_42^6 + 5*z_42^4 + 5*z_42^3 - 13*z_42 + 3)*x_5^2 + 1/2*(11*z_42^11 - 17*z_42^8 + 17*z_42^6 - 11*z_42^4 - 11*z_42^3 + 17*z_42 - 2)*x_5*x_6 + 1/2*(-z_42^11 + 5*z_42^8 - 5*z_42^6 + z_42^4 + z_42^3 - 5*z_42 - 1)*x_6^2,
x_0*x_3 + 1/4*(z_42^11 - 2*z_42^8 + 2*z_42^6 - z_42^4 - z_42^3 + 2*z_42 - 2)*x_1*x_3 + 1/4*(5*z_42^11 - 11*z_42^8 + 11*z_42^6 - 5*z_42^4 - 5*z_42^3 + 11*z_42 - 5)*x_1*x_4 + 1/4*(-4*z_42^11 + 13*z_42^8 - 13*z_42^6 + 4*z_42^4 + 4*z_42^3 - 13*z_42 + 5)*x_1*x_5 + 1/4*(11*z_42^11 - 19*z_42^8 + 19*z_42^6 - 11*z_42^4 - 11*z_42^3 + 19*z_42 - 3)*x_1*x_6 + 1/2*(-z_42^11 + 2*z_42^8 - 2*z_42^6 + z_42^4 + z_42^3 - 2*z_42 + 1)*x_2^2 + 1/4*(3*z_42^11 - 4*z_42^8 + 4*z_42^6 - 3*z_42^4 - 3*z_42^3 + 4*z_42 + 5)*x_2*x_3 + 1/4*(z_42^11 - 2*z_42^8 + 2*z_42^6 - z_42^4 - z_42^3 + 2*z_42)*x_2*x_4 + 1/4*(-2*z_42^11 + 2*z_42^4 + 2*z_42^3 + 1)*x_2*x_5 + 1/4*(-13*z_42^11 + 20*z_42^8 - 20*z_42^6 + 13*z_42^4 + 13*z_42^3 - 20*z_42 + 2)*x_2*x_6 + 1/4*(-5*z_42^11 + 7*z_42^8 - 7*z_42^6 + 5*z_42^4 + 5*z_42^3 - 7*z_42 + 2)*x_3^2 + 1/4*(-3*z_42^11 + 4*z_42^8 - 4*z_42^6 + 3*z_42^4 + 3*z_42^3 - 4*z_42 + 2)*x_3*x_4 + 1/4*(9*z_42^11 - 19*z_42^8 + 19*z_42^6 - 9*z_42^4 - 9*z_42^3 + 19*z_42 - 2)*x_3*x_5 + 1/4*(-z_42^11 + 3*z_42^8 - 3*z_42^6 + z_42^4 + z_42^3 - 3*z_42)*x_3*x_6 + 1/4*(-2*z_42^11 + 5*z_42^8 - 5*z_42^6 + 2*z_42^4 + 2*z_42^3 - 5*z_42 + 4)*x_4^2 + 1/4*(8*z_42^11 - 18*z_42^8 + 18*z_42^6 - 8*z_42^4 - 8*z_42^3 + 18*z_42 - 5)*x_4*x_5 + 1/4*(-5*z_42^11 + 10*z_42^8 - 10*z_42^6 + 5*z_42^4 + 5*z_42^3 - 10*z_42 + 5)*x_4*x_6 + 1/4*(z_42^11 - 5*z_42^8 + 5*z_42^6 - z_42^4 - z_42^3 + 5*z_42 - 3)*x_5^2 + 1/4*(-5*z_42^11 + 7*z_42^8 - 7*z_42^6 + 5*z_42^4 + 5*z_42^3 - 7*z_42)*x_5*x_6 + 1/4*(z_42^11 - z_42^8 + z_42^6 - z_42^4 - z_42^3 + z_42 - 1)*x_6^2,
x_0*x_4 + 1/2*(-9*z_42^11 + 12*z_42^8 - 12*z_42^6 + 9*z_42^4 + 9*z_42^3 - 12*z_42 + 7)*x_1*x_3 + 1/2*(-23*z_42^11 + 42*z_42^8 - 42*z_42^6 + 23*z_42^4 + 23*z_42^3 - 42*z_42 + 12)*x_1*x_4 + 1/2*(20*z_42^11 - 40*z_42^8 + 40*z_42^6 - 20*z_42^4 - 20*z_42^3 + 40*z_42 - 9)*x_1*x_5 + 1/2*(-31*z_42^11 + 56*z_42^8 - 56*z_42^6 + 31*z_42^4 + 31*z_42^3 - 56*z_42 + 14)*x_1*x_6 + (4*z_42^11 - 7*z_42^8 + 7*z_42^6 - 4*z_42^4 - 4*z_42^3 + 7*z_42 - 2)*x_2^2 + 1/2*(-2*z_42^11 + 7*z_42^8 - 7*z_42^6 + 2*z_42^4 + 2*z_42^3 - 7*z_42 - 2)*x_2*x_3 + 1/2*(-z_42^11 + 2*z_42^8 - 2*z_42^6 + z_42^4 + z_42^3 - 2*z_42 + 1)*x_2*x_4 + 1/2*(z_42^11 - z_42^8 + z_42^6 - z_42^4 - z_42^3 + z_42 - 1)*x_2*x_5 + 1/2*(33*z_42^11 - 58*z_42^8 + 58*z_42^6 - 33*z_42^4 - 33*z_42^3 + 58*z_42 - 13)*x_2*x_6 + 1/2*(12*z_42^11 - 23*z_42^8 + 23*z_42^6 - 12*z_42^4 - 12*z_42^3 + 23*z_42 - 5)*x_3^2 + 1/2*(9*z_42^11 - 16*z_42^8 + 16*z_42^6 - 9*z_42^4 - 9*z_42^3 + 16*z_42 - 5)*x_3*x_4 + 1/2*(-28*z_42^11 + 53*z_42^8 - 53*z_42^6 + 28*z_42^4 + 28*z_42^3 - 53*z_42 + 11)*x_3*x_5 + 1/2*(6*z_42^11 - 7*z_42^8 + 7*z_42^6 - 6*z_42^4 - 6*z_42^3 + 7*z_42 - 5)*x_3*x_6 + 1/2*(11*z_42^11 - 21*z_42^8 + 21*z_42^6 - 11*z_42^4 - 11*z_42^3 + 21*z_42 - 6)*x_4^2 + 1/2*(-27*z_42^11 + 49*z_42^8 - 49*z_42^6 + 27*z_42^4 + 27*z_42^3 - 49*z_42 + 13)*x_4*x_5 + 1/2*(20*z_42^11 - 35*z_42^8 + 35*z_42^6 - 20*z_42^4 - 20*z_42^3 + 35*z_42 - 12)*x_4*x_6 + 1/2*(-9*z_42^11 + 18*z_42^8 - 18*z_42^6 + 9*z_42^4 + 9*z_42^3 - 18*z_42 + 4)*x_5^2 + 1/2*(14*z_42^11 - 23*z_42^8 + 23*z_42^6 - 14*z_42^4 - 14*z_42^3 + 23*z_42 - 5)*x_5*x_6 + 1/2*(-3*z_42^11 + 6*z_42^8 - 6*z_42^6 + 3*z_42^4 + 3*z_42^3 - 6*z_42)*x_6^2,
x_0*x_5 + 1/4*(-11*z_42^11 + 22*z_42^8 - 22*z_42^6 + 11*z_42^4 + 11*z_42^3 - 22*z_42 + 4)*x_1*x_3 + 1/4*(-35*z_42^11 + 63*z_42^8 - 63*z_42^6 + 35*z_42^4 + 35*z_42^3 - 63*z_42 + 13)*x_1*x_4 + 1/4*(32*z_42^11 - 57*z_42^8 + 57*z_42^6 - 32*z_42^4 - 32*z_42^3 + 57*z_42 - 15)*x_1*x_5 + 1/4*(-45*z_42^11 + 83*z_42^8 - 83*z_42^6 + 45*z_42^4 + 45*z_42^3 - 83*z_42 + 19)*x_1*x_6 + 1/2*(5*z_42^11 - 10*z_42^8 + 10*z_42^6 - 5*z_42^4 - 5*z_42^3 + 10*z_42 - 3)*x_2^2 + 1/4*(-3*z_42^11 + 6*z_42^8 - 6*z_42^6 + 3*z_42^4 + 3*z_42^3 - 6*z_42 + 3)*x_2*x_3 + 1/4*(z_42^11 + 2*z_42^8 - 2*z_42^6 - z_42^4 - z_42^3 - 2*z_42 + 2)*x_2*x_4 + 1/4*(-2*z_42^8 + 2*z_42^6 + 2*z_42 + 1)*x_2*x_5 + 1/4*(47*z_42^11 - 84*z_42^8 + 84*z_42^6 - 47*z_42^4 - 47*z_42^3 + 84*z_42 - 20)*x_2*x_6 + 1/4*(21*z_42^11 - 33*z_42^8 + 33*z_42^6 - 21*z_42^4 - 21*z_42^3 + 33*z_42 - 8)*x_3^2 + 1/4*(13*z_42^11 - 24*z_42^8 + 24*z_42^6 - 13*z_42^4 - 13*z_42^3 + 24*z_42 - 8)*x_3*x_4 + 1/4*(-41*z_42^11 + 73*z_42^8 - 73*z_42^6 + 41*z_42^4 + 41*z_42^3 - 73*z_42 + 20)*x_3*x_5 + 1/4*(5*z_42^11 - 13*z_42^8 + 13*z_42^6 - 5*z_42^4 - 5*z_42^3 + 13*z_42 - 2)*x_3*x_6 + 1/4*(16*z_42^11 - 31*z_42^8 + 31*z_42^6 - 16*z_42^4 - 16*z_42^3 + 31*z_42 - 8)*x_4^2 + 1/4*(-38*z_42^11 + 72*z_42^8 - 72*z_42^6 + 38*z_42^4 + 38*z_42^3 - 72*z_42 + 19)*x_4*x_5 + 1/4*(29*z_42^11 - 56*z_42^8 + 56*z_42^6 - 29*z_42^4 - 29*z_42^3 + 56*z_42 - 9)*x_4*x_6 + 1/4*(-15*z_42^11 + 25*z_42^8 - 25*z_42^6 + 15*z_42^4 + 15*z_42^3 - 25*z_42 + 7)*x_5^2 + 1/4*(17*z_42^11 - 33*z_42^8 + 33*z_42^6 - 17*z_42^4 - 17*z_42^3 + 33*z_42 - 10)*x_5*x_6 + 1/4*(-3*z_42^11 + 5*z_42^8 - 5*z_42^6 + 3*z_42^4 + 3*z_42^3 - 5*z_42 + 5)*x_6^2,
x_0*x_6 + 1/4*(5*z_42^11 + 4*z_42^8 - 4*z_42^6 - 5*z_42^4 - 5*z_42^3 - 4*z_42)*x_1*x_3 + 1/4*(-z_42^11 + 3*z_42^8 - 3*z_42^6 + z_42^4 + z_42^3 - 3*z_42 - 1)*x_1*x_4 + 1/4*(-2*z_42^11 + z_42^8 - z_42^6 + 2*z_42^4 + 2*z_42^3 - z_42 - 5)*x_1*x_5 + 1/4*(-3*z_42^11 + 3*z_42^8 - 3*z_42^6 + 3*z_42^4 + 3*z_42^3 - 3*z_42 - 3)*x_1*x_6 + 1/2*(-z_42^11 + z_42^4 + z_42^3 - 1)*x_2^2 + 1/4*(-5*z_42^11 + 5*z_42^4 + 5*z_42^3 - 1)*x_2*x_3 + 1/4*(z_42^11 - z_42^4 - z_42^3 + 2)*x_2*x_4 + 1/4*(-2*z_42^11 + 2*z_42^8 - 2*z_42^6 + 2*z_42^4 + 2*z_42^3 - 2*z_42 - 3)*x_2*x_5 + 1/4*(3*z_42^11 - 2*z_42^8 + 2*z_42^6 - 3*z_42^4 - 3*z_42^3 + 2*z_42 + 4)*x_2*x_6 + 1/4*(5*z_42^11 - z_42^8 + z_42^6 - 5*z_42^4 - 5*z_42^3 + z_42 + 2)*x_3^2 + 1/4*(z_42^11 - 2*z_42^8 + 2*z_42^6 - z_42^4 - z_42^3 + 2*z_42)*x_3*x_4 + 1/4*(-5*z_42^11 + z_42^8 - z_42^6 + 5*z_42^4 + 5*z_42^3 - z_42 - 2)*x_3*x_5 + 1/4*(-3*z_42^11 - 5*z_42^8 + 5*z_42^6 + 3*z_42^4 + 3*z_42^3 + 5*z_42 + 4)*x_3*x_6 + 1/4*(-z_42^8 + z_42^6 + z_42 - 2)*x_4^2 + 1/4*(4*z_42^11 - 4*z_42^4 - 4*z_42^3 + 3)*x_4*x_5 + 1/4*(3*z_42^11 - 6*z_42^8 + 6*z_42^6 - 3*z_42^4 - 3*z_42^3 + 6*z_42 + 7)*x_4*x_6 + 1/4*(-z_42^11 + z_42^8 - z_42^6 + z_42^4 + z_42^3 - z_42 + 1)*x_5^2 + 1/4*(z_42^11 - z_42^8 + z_42^6 - z_42^4 - z_42^3 + z_42)*x_5*x_6 + 1/4*(3*z_42^11 - 3*z_42^8 + 3*z_42^6 - 3*z_42^4 - 3*z_42^3 + 3*z_42 + 7)*x_6^2,
x_1^2 + (-6*z_42^11 + 14*z_42^8 - 14*z_42^6 + 6*z_42^4 + 6*z_42^3 - 14*z_42 - 2)*x_1*x_3 + (-18*z_42^11 + 34*z_42^8 - 34*z_42^6 + 18*z_42^4 + 18*z_42^3 - 34*z_42 + 6)*x_1*x_4 + (18*z_42^11 - 30*z_42^8 + 30*z_42^6 - 18*z_42^4 - 18*z_42^3 + 30*z_42 - 8)*x_1*x_5 + (-24*z_42^11 + 44*z_42^8 - 44*z_42^6 + 24*z_42^4 + 24*z_42^3 - 44*z_42 + 10)*x_1*x_6 + (7*z_42^11 - 12*z_42^8 + 12*z_42^6 - 7*z_42^4 - 7*z_42^3 + 12*z_42 - 2)*x_2^2 + (-2*z_42^11 + 2*z_42^8 - 2*z_42^6 + 2*z_42^4 + 2*z_42^3 - 2*z_42 + 4)*x_2*x_3 + (-2*z_42^11 + 2*z_42^8 - 2*z_42^6 + 2*z_42^4 + 2*z_42^3 - 2*z_42)*x_2*x_4 + (2*z_42^11 - 2*z_42^8 + 2*z_42^6 - 2*z_42^4 - 2*z_42^3 + 2*z_42)*x_2*x_5 + (24*z_42^11 - 44*z_42^8 + 44*z_42^6 - 24*z_42^4 - 24*z_42^3 + 44*z_42 - 12)*x_2*x_6 + (10*z_42^11 - 17*z_42^8 + 17*z_42^6 - 10*z_42^4 - 10*z_42^3 + 17*z_42 - 7)*x_3^2 + (8*z_42^11 - 14*z_42^8 + 14*z_42^6 - 8*z_42^4 - 8*z_42^3 + 14*z_42 - 2)*x_3*x_4 + (-22*z_42^11 + 40*z_42^8 - 40*z_42^6 + 22*z_42^4 + 22*z_42^3 - 40*z_42 + 12)*x_3*x_5 + (2*z_42^11 - 10*z_42^8 + 10*z_42^6 - 2*z_42^4 - 2*z_42^3 + 10*z_42 + 2)*x_3*x_6 + (10*z_42^11 - 17*z_42^8 + 17*z_42^6 - 10*z_42^4 - 10*z_42^3 + 17*z_42 - 3)*x_4^2 + (-22*z_42^11 + 40*z_42^8 - 40*z_42^6 + 22*z_42^4 + 22*z_42^3 - 40*z_42 + 8)*x_4*x_5 + (14*z_42^11 - 30*z_42^8 + 30*z_42^6 - 14*z_42^4 - 14*z_42^3 + 30*z_42 - 6)*x_4*x_6 + (-8*z_42^11 + 13*z_42^8 - 13*z_42^6 + 8*z_42^4 + 8*z_42^3 - 13*z_42 + 4)*x_5^2 + (10*z_42^11 - 18*z_42^8 + 18*z_42^6 - 10*z_42^4 - 10*z_42^3 + 18*z_42 - 4)*x_5*x_6 + (-4*z_42^11 + 4*z_42^8 - 4*z_42^6 + 4*z_42^4 + 4*z_42^3 - 4*z_42 + 1)*x_6^2,
x_1*x_2 + (-3*z_42^11 + 6*z_42^8 - 6*z_42^6 + 3*z_42^4 + 3*z_42^3 - 6*z_42 - 1)*x_1*x_3 + (-7*z_42^11 + 14*z_42^8 - 14*z_42^6 + 7*z_42^4 + 7*z_42^3 - 14*z_42 + 3)*x_1*x_4 + (7*z_42^11 - 12*z_42^8 + 12*z_42^6 - 7*z_42^4 - 7*z_42^3 + 12*z_42 - 3)*x_1*x_5 + (-10*z_42^11 + 18*z_42^8 - 18*z_42^6 + 10*z_42^4 + 10*z_42^3 - 18*z_42 + 4)*x_1*x_6 + (3*z_42^11 - 5*z_42^8 + 5*z_42^6 - 3*z_42^4 - 3*z_42^3 + 5*z_42 - 1)*x_2^2 + (-z_42^11 + z_42^8 - z_42^6 + z_42^4 + z_42^3 - z_42 + 1)*x_2*x_3 + (-z_42^11 + z_42^8 - z_42^6 + z_42^4 + z_42^3 - z_42 - 1)*x_2*x_4 + (z_42^11 - z_42^8 + z_42^6 - z_42^4 - z_42^3 + z_42)*x_2*x_5 + (10*z_42^11 - 18*z_42^8 + 18*z_42^6 - 10*z_42^4 - 10*z_42^3 + 18*z_42 - 5)*x_2*x_6 + (4*z_42^11 - 7*z_42^8 + 7*z_42^6 - 4*z_42^4 - 4*z_42^3 + 7*z_42 - 3)*x_3^2 + (4*z_42^11 - 6*z_42^8 + 6*z_42^6 - 4*z_42^4 - 4*z_42^3 + 6*z_42)*x_3*x_4 + (-9*z_42^11 + 16*z_42^8 - 16*z_42^6 + 9*z_42^4 + 9*z_42^3 - 16*z_42 + 5)*x_3*x_5 + (z_42^11 - 4*z_42^8 + 4*z_42^6 - z_42^4 - z_42^3 + 4*z_42 + 1)*x_3*x_6 + (4*z_42^11 - 7*z_42^8 + 7*z_42^6 - 4*z_42^4 - 4*z_42^3 + 7*z_42 - 1)*x_4^2 + (-9*z_42^11 + 16*z_42^8 - 16*z_42^6 + 9*z_42^4 + 9*z_42^3 - 16*z_42 + 3)*x_4*x_5 + (5*z_42^11 - 12*z_42^8 + 12*z_42^6 - 5*z_42^4 - 5*z_42^3 + 12*z_42 - 3)*x_4*x_6 + (-3*z_42^11 + 5*z_42^8 - 5*z_42^6 + 3*z_42^4 + 3*z_42^3 - 5*z_42 + 2)*x_5^2 + (5*z_42^11 - 8*z_42^8 + 8*z_42^6 - 5*z_42^4 - 5*z_42^3 + 8*z_42 - 1)*x_5*x_6 + (-2*z_42^11 + 2*z_42^8 - 2*z_42^6 + 2*z_42^4 + 2*z_42^3 - 2*z_42)*x_6^2
});
o5 : Ideal of T
i6 : betti res I
0 1 2 3 4 5
o6 = total: 1 10 16 16 10 1
0: 1 . . . . .
1: . 10 16 . . .
2: . . . 16 10 .
3: . . . . . 1
o6 : BettiTally