Finding equations of a 1-parameter family of genus 7 Riemann surfaces with 32 automorphisms

Magaard, Shaska, Shpectorov, and Völklein give tables of smooth Riemann surfaces of genus \( 3 \leq g \leq 10\) with automorphism groups \(G\) satisfying \( \# G > 4(g-1)\). Their list is based on a computer search by Breuer.

They list a family of genus 7 Riemann surfaces with automorphism group (32,42) in the GAP library of small groups. The quotient of any member of this family by its automorphism group has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,2,8).

We use GAP, Macaulay2 and Magma to compute equations of one curve in this pencil, and give a conjectural description of this pencil.

Obtaining candidate polynomials in `Magma`

We use some Magma code developed by David Swinarski during a visit to the University of Sydney in June/July 2011. Here is the file autcv10.txt used below.

Magma V2.21-10    Sun Mar 13 2016 22:10:56 on ace-math01 [Seed = 4182356564]
Type ? for help.  Type -D to quit.
> load "autcv10c.txt";
Loading "autcv10c.txt"
> G:=SmallGroup(32,42);
> MatrixG,MatrixSKG,Q,C:=RunExample(G,7,[2,2,2,8]);
Set seed to 0.


Character Table of Group G
--------------------------


----------------------------------------------------------
Class |   1  2  3  4  5    6    7  8  9 10  11  12  13  14
Size  |   1  1  2  4  4    1    1  2  4  4   2   2   2   2
Order |   1  2  2  2  2    4    4  4  4  4   8   8   8   8
----------------------------------------------------------
p  =  2   1  1  1  1  1    2    2  2  2  2   8   8   8   8
----------------------------------------------------------
X.1   +   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  -1
X.3   +   1  1 -1  1  1   -1   -1  1 -1 -1  -1  -1   1   1
X.4   +   1  1 -1  1 -1   -1   -1  1  1 -1   1   1  -1  -1
X.5   +   1  1 -1 -1  1   -1   -1  1 -1  1   1   1  -1  -1
X.6   +   1  1 -1 -1 -1   -1   -1  1  1  1  -1  -1   1   1
X.7   +   1  1  1 -1  1    1    1  1  1 -1  -1  -1  -1  -1
X.8   +   1  1  1 -1 -1    1    1  1 -1 -1   1   1   1   1
X.9   +   2  2 -2  0  0    2    2 -2  0  0   0   0   0   0
X.10  +   2  2  2  0  0   -2   -2 -2  0  0   0   0   0   0
X.11  0   2 -2  0  0  0 -2*I  2*I  0  0  0  Z1 -Z1  Z2 -Z2
X.12  0   2 -2  0  0  0 -2*I  2*I  0  0  0 -Z1  Z1 -Z2  Z2
X.13  0   2 -2  0  0  0  2*I -2*I  0  0  0 -Z1  Z1  Z2 -Z2
X.14  0   2 -2  0  0  0  2*I -2*I  0  0  0  Z1 -Z1 -Z2  Z2


Explanation of Character Value Symbols
--------------------------------------

# denotes algebraic conjugation, that is,
#k indicates replacing the root of unity w by w^k

I = RootOfUnity(4)

Z1     = (CyclotomicField(8: Sparse := true)) ! [ RationalField() | 0, -1, 0, -1
]

Z2     = (CyclotomicField(8: Sparse := true)) ! [ RationalField() | 0, -1, 0, 1 
]


Conjugacy Classes of group G
----------------------------
[1]     Order 1       Length 1      
        Rep Id(G)

[2]     Order 2       Length 1      
        Rep G.5

[3]     Order 2       Length 2      
        Rep G.3 * G.4

[4]     Order 2       Length 4      
        Rep G.2

[5]     Order 2       Length 4      
        Rep G.1

[6]     Order 4       Length 1      
        Rep G.3 * G.5

[7]     Order 4       Length 1      
        Rep G.3

[8]     Order 4       Length 2      
        Rep G.4

[9]     Order 4       Length 4      
        Rep G.1 * G.3

[10]    Order 4       Length 4      
        Rep G.2 * G.3

[11]    Order 8       Length 2      
        Rep G.1 * G.2 * G.3 * G.5

[12]    Order 8       Length 2      
        Rep G.1 * G.2 * G.3

[13]    Order 8       Length 2      
        Rep G.1 * G.2 * G.5

[14]    Order 8       Length 2      
        Rep G.1 * G.2


Surface kernel generators:  [ G.2 * G.4, G.1 * G.4 * G.5, G.3 * G.4 * G.5, G.1 *
G.2 * G.3 * 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, 0 ]
S_1      =[ 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1 ]
I_2      =[ 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0 ]
S_2      =[ 2, 2, 2, 0, 0, 0, 2, 0, 1, 3, 1, 2, 2, 1 ]
H^0(C,2K)=[ 1, 1, 1, 0, 0, 0, 1, 0, 1, 2, 1, 1, 1, 1 ]
I_3      =[ 1, 0, 1, 3, 3, 2, 0, 2, 5, 2, 3, 4, 4, 3 ]
S_3      =[ 1, 1, 2, 4, 4, 4, 1, 3, 7, 3, 5, 6, 6, 5 ]
H^0(C,3K)=[ 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2 ]
I2timesS1=[ 2, 0, 1, 4, 4, 2, 0, 3, 6, 3, 4, 5, 5, 4 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 32 and degree 16
[
    [  -1    0    0    0    0    0    0]
    [   0   -1    0    0    0    0    0]
    [   0    0    1    0    0    0    0]
    [   0    0    0    0    1    0    0]
    [   0    0    0    1    0    0    0]
    [   0    0    0    0    0    0 -z^8]
    [   0    0    0    0    0  z^8    0],

    [   -1     0     0     0     0     0     0]
    [    0     0     1     0     0     0     0]
    [    0     1     0     0     0     0     0]
    [    0     0     0     0 -z^12     0     0]
    [    0     0     0   z^4     0     0     0]
    [    0     0     0     0     0     0   z^4]
    [    0     0     0     0     0 -z^12     0],

    [  -1    0    0    0    0    0    0]
    [   0    1    0    0    0    0    0]
    [   0    0    1    0    0    0    0]
    [   0    0    0  z^8    0    0    0]
    [   0    0    0    0  z^8    0    0]
    [   0    0    0    0    0 -z^8    0]
    [   0    0    0    0    0    0 -z^8]
]
Matrix Surface Kernel Generators:
[
    [   -1     0     0     0     0     0     0]
    [    0     0    -1     0     0     0     0]
    [    0    -1     0     0     0     0     0]
    [    0     0     0     0   z^4     0     0]
    [    0     0     0 -z^12     0     0     0]
    [    0     0     0     0     0     0  z^12]
    [    0     0     0     0     0  -z^4     0],

    [  -1    0    0    0    0    0    0]
    [   0    1    0    0    0    0    0]
    [   0    0   -1    0    0    0    0]
    [   0    0    0    0 -z^8    0    0]
    [   0    0    0  z^8    0    0    0]
    [   0    0    0    0    0    0   -1]
    [   0    0    0    0    0   -1    0],

    [-1  0  0  0  0  0  0]
    [ 0 -1  0  0  0  0  0]
    [ 0  0 -1  0  0  0  0]
    [ 0  0  0 -1  0  0  0]
    [ 0  0  0  0  1  0  0]
    [ 0  0  0  0  0  1  0]
    [ 0  0  0  0  0  0 -1],

    [  -1    0    0    0    0    0    0]
    [   0    0    1    0    0    0    0]
    [   0   -1    0    0    0    0    0]
    [   0    0    0  z^4    0    0    0]
    [   0    0    0    0 z^12    0    0]
    [   0    0    0    0    0  z^4    0]
    [   0    0    0    0    0    0 z^12]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 2
Multiplicity 2
[
    x_0^2,
    x_1^2 + x_2^2
]
I2 contains a 1-dimensional subspace of CharacterRow 2
Dimension 2
Multiplicity 2
[
    x_1*x_2,
    x_3*x_6 + z^8*x_4*x_5
]
I2 contains a 1-dimensional subspace of CharacterRow 3
Dimension 2
Multiplicity 2
[
    x_3*x_4,
    x_5*x_6
]
I2 contains a 1-dimensional subspace of CharacterRow 7
Dimension 2
Multiplicity 2
[
    x_1^2 - x_2^2,
    x_3*x_6 - z^8*x_4*x_5
]
I2 contains a 2-dimensional subspace of CharacterRow 10
Dimension 6
Multiplicity 3
[
    x_0*x_1,
    x_0*x_2,
    x_3^2,
    x_4^2,
    x_5^2,
    x_6^2
]
I2 contains a 2-dimensional subspace of CharacterRow 12
Dimension 4
Multiplicity 2
[
    x_0*x_5,
    x_0*x_6,
    x_1*x_3 - z^8*x_2*x_3,
    x_1*x_4 + z^8*x_2*x_4
]
I2 contains a 2-dimensional subspace of CharacterRow 13
Dimension 4
Multiplicity 2
[
    x_0*x_3,
    x_0*x_4,
    x_1*x_5 - z^8*x_2*x_5,
    x_1*x_6 + z^8*x_2*x_6
]

The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields a polynomial of the form


c_1*(x_0^2)+c_2*(x_1^2 + x_2^2)

Assume that \((c_1,c_2)\) are nonzero. After scaling \(x_0\), we may assume that \(c_1 = c_2=1.\)

The second isotypical subspace, which corresponds to the character \( \chi_2\) in the character table shown above, yields a polynomial of the form


c_3*(x_1*x_2)+c_4*(x_3*x_6 + i*x_4*x_5)

Assume that \(c_3,c_4\) are nonzero. After scaling \(x_1,x_2\) and dividing, we may assume that \(c_3=c_4=1\).

The third isotypical subspace, which corresponds to the character \( \chi_3\) in the character table shown above, yields a polynomial of the form


c_5*(x_3*x_4)+c_6*(x_5*x_6)

Assume that \(c_5,c_6 \) are nonzero. After scaling \(x_3,x_4\) and dividing, we may assume that \(c_5 = c_6= 1.\)

The fourth isotypical subspace, which corresponds to the character \( \chi_7\) in the character table shown above, yields a polynomial of the form


c_7*(x_1^2 - x_2^2)+c_8*(x_3*x_6 - i*x_4*x_5)

Assume that \(c_7\) is nonzero. After dividing by \(c_7\), we may assume that \(c_7= 1.\)

The fifth isotypical subspace corresponds to the character \( \chi_{10}\) in the character table shown above. Note that the matrix surface kernel generators have a block diagonal form with blocks of size \(1 \times 1\), \(2 \times 2\), \(2 \times 2\), and \(2 \times 2\). We therefore let the first two polynomials shown here generate one copy of \(V_{10}\) and find two ordered bases of the span of the last four polynomials such that the action of \(G\) is given by the same matrices relative to all three ordered bases.


> GL7K:=Parent(MatrixG[1]);
> MatrixG:=sub<GL7K | MatrixG>;
> FindParallelBases(MatrixG, [Q[5][1],Q[5][2]],[Q[5][3],Q[5][4]]); 
[        x_3^2 + x_4^2]
[z^8*x_3^2 - z^8*x_4^2]
> FindParallelBases(MatrixG, [Q[5][1],Q[5][2]],[Q[5][5],Q[5][6]]); 
[        x_5^2 - x_6^2]
[z^8*x_5^2 + z^8*x_6^2]

This yields the polynomials


    c_9*(x_0*x_1) + c_10*(x_3^2 + x_4^2) + c_11*(x_5^2 - x_6^2),
    c_9*(x_0*x_2) + c_10*(i*x_3^2 - i*x_4^2) + c_11*(i*x_5^2 + i*x_6^2),

Assume that \(c_{9}\) is nonzero. After dividing by \(c_9\), we may assume that \(c_{9} = 1.\)

The sixth isotypical subspace corresponds to the character \( \chi_{12}\) in the character table shown above. Again we find two ordered bases such that the action of \(G\) is given by the same matrices relative to both ordered bases. This yields the polynomials


c_12*(x_0*x_5)+c_13*(i*x_1*x_4 - x_2*x_4),
c_12*(x_0*x_6)+c_13*(x_1*x_3 - i*x_2*x_3).

Assume that \(c_{12}\) is nonzero. After dividing by \(c_{12}\), we may assume that \(c_{12} = 1.\)

The seventh isotypical subspace corresponds to the character \( \chi_{13}\) in the character table shown above. Again we find two ordered bases such that the action of \(G\) is given by the same matrices relative to both ordered bases. This yields the polynomials


c_14*(x_0*x_3)+c_15*(i*x_1*x_6 - x_2*x_6),
c_14*(x_0*x_4)+c_15*(x_1*x_5 - i*x_2*x_5).

Assume that \(c_{14}\) is nonzero. After dividing by \(c_{14}\), we may assume that \(c_{14} = 1.\)

We collect all the polynomials listed above and the assumptions we made about the coefficients \(c_i\) to obtain the following candidate polynomials: \[ \begin{array}{l} x_0^2+x_1^2 + x_2^2,\\ x_1 x_2+x_3 x_6 + i x_4 x_5,\\ x_3 x_4+ x_5 x_6,\\ x_1^2 - x_2^2+c_8 (x_3 x_6 - i x_4 x_5),\\ x_0 x_1+c_{10} (x_3^2 + x_4^2)+c_{11} (x_5^2 -x_6^2),\\ x_0 x_2+c_{10} (i x_3^2 - i x_4^2)+c_{11} (i x_5^2 + i x_6^2),\\ x_0 x_5+c_{13} (i x_1 x_4 - x_2 x_4),\\ x_0 x_6+c_{13} (x_1 x_3 -i x_2 x_3),\\ x_0 x_3+c_{15} (i x_1 x_6 - x_2 x_6),\\ x_0 x_4+c_{15} (x_1 x_5 -i x_2 x_5) \end{array} \] For generic values of \(c_8, c_{10},c_{11},c_{13},c_{15}\), it appears that the intersection of these 10 quadrics in \(\mathbb{P}^6\) is zero-dimensional. Here is an example showing this:


> K<i>:=CyclotomicField(4);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_8:=1;
> c_10:=1;
> c_11:=1;
> c_13:=1;
> c_15:=1;
> X:=Scheme(P6,[
> x_0^2+x_1^2 + x_2^2,
> x_1*x_2+x_3*x_6 + i*x_4*x_5,
> x_3*x_4+x_5*x_6,
> x_1^2 - x_2^2+c_8*(x_3*x_6 - i*x_4*x_5),
> x_0*x_1+c_10*(x_3^2 + x_4^2)+c_11*(x_5^2 -x_6^2),
> x_0*x_2+c_10*(i*x_3^2 - i*x_4^2)+c_11*(i*x_5^2 + i*x_6^2),
> x_0*x_5+c_13*(i*x_1*x_4 - x_2*x_4),
> x_0*x_6+c_13*(x_1*x_3 -i*x_2*x_3),
> x_0*x_3+c_15*(i*x_1*x_6 - x_2*x_6),
> x_0*x_4+c_15*(x_1*x_5 -i*x_2*x_5)
> ]);
> Dimension(X);
0

Therefore next we turn to Macaulay2 to compute part of a flattening stratification. (We switch software packages because, to the best of our knowledge, Magma will not compute Gröbner bases in a polynomial ring over a polynomial ring.)

Flattening stratification in `Macaulay2`

We compute the degree two and three elements in a Gröbner basis in Macaulay2 for the ideal generated by the candidate polynomials.

Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(4);

i3 : i=K_0;

i4 : S=K[c_8,c_10,c_11,c_13,c_15,Degrees=>{0,0,0,0,0}];

i5 : T=S[x_0..x_6];

i6 : I=ideal({
     x_0^2+x_1^2 + x_2^2,
     x_1*x_2+x_3*x_6 + i*x_4*x_5,
     x_3*x_4+x_5*x_6,
     x_1^2 - x_2^2+c_8*(x_3*x_6 - i*x_4*x_5),
     x_0*x_1+c_10*(x_3^2 + x_4^2)+c_11*(x_5^2 -x_6^2),
     x_0*x_2+c_10*(i*x_3^2 - i*x_4^2)+c_11*(i*x_5^2 + i*x_6^2),
     x_0*x_5+c_13*(i*x_1*x_4 - x_2*x_4),
     x_0*x_6+c_13*(x_1*x_3 -i*x_2*x_3),
     x_0*x_3+c_15*(i*x_1*x_6 - x_2*x_6),
     x_0*x_4+c_15*(x_1*x_5 -i*x_2*x_5)
     });

o6 : Ideal of T

i7 : L=flatten entries gens gb(I,DegreeLimit=>6);

i8 : L3=select(L, i -> degree i == {3,0});

i9 : L3c=unique apply(L3, i -> leadCoefficient i);

i10 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
c_8*c_15-2*c_10-2*ww_4*c_15
c_13+c_15
c_10*c_13*c_15-ww_4*c_10
c_15
c_8*c_13+2*c_10-2*ww_4*c_13
c_13*c_15-ww_4
c_8*c_15+2*ww_4*c_11+2*ww_4*c_15
c_11*c_13+ww_4*c_10*c_15+2*c_13*c_15
c_8*c_13*c_15+2*c_10*c_15-2*ww_4*c_13*c_15
c_11*c_13+(1/2)*c_8+ww_4
c_10*c_13*c_15+(1/2)*ww_4*c_8*c_13+c_13
c_8+2*ww_4
c_10*c_15+2
c_10
1

This suggests setting \(c_{15} =-c_{13}\). We repeat the calculation with this choice:

i11 : S=K[c_8,c_10,c_11,c_13,Degrees=>{0,0,0,0}];

i12 : c_15=-c_13;

i13 : T=S[x_0..x_6];

i14 : I=ideal({
      x_0^2+x_1^2 + x_2^2,
      x_1*x_2+x_3*x_6 + i*x_4*x_5,
      x_3*x_4+x_5*x_6,
      x_1^2 - x_2^2+c_8*(x_3*x_6 - i*x_4*x_5),
      x_0*x_1+c_10*(x_3^2 + x_4^2)+c_11*(x_5^2 -x_6^2),
      x_0*x_2+c_10*(i*x_3^2 - i*x_4^2)+c_11*(i*x_5^2 + i*x_6^2),
      x_0*x_5+c_13*(i*x_1*x_4 - x_2*x_4),
      x_0*x_6+c_13*(x_1*x_3 -i*x_2*x_3),
      x_0*x_3+c_15*(i*x_1*x_6 - x_2*x_6),
      x_0*x_4+c_15*(x_1*x_5 -i*x_2*x_5)
      });

o14 : Ideal of T

i15 : L=flatten entries gens gb(I,DegreeLimit=>6);

i16 : L3=select(L, i -> degree i == {3,0});

i17 : L3c=unique apply(L3, i -> leadCoefficient i);

i18 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
c_8*c_13+2*c_10-2*ww_4*c_13
c_10*c_13^2+ww_4*c_10
c_13^2+ww_4
c_8*c_13-2*ww_4*c_11+2*ww_4*c_13
c_10*c_13+ww_4*c_11*c_13-2*ww_4*c_13^2
c_8*c_13^2-2*ww_4*c_11*c_13+2*ww_4*c_13^2
c_10*c_13^2-(1/2)*ww_4*c_8*c_13-c_13
c_8*c_10*c_13-(1/2)*ww_4*c_8^2+2*ww_4*c_10*c_13-2*ww_4
1
c_10

This suggests setting \( c_{13}^2+i =0\), or \(c_{13} = \zeta_{8}^3.\) We repeat the calculation with this choice:

i19 : K=cyclotomicField(8);

i20 : z_8=K_0;

i21 : i=z_8^2;

i22 : S=K[c_8,c_10,c_11,Degrees=>{0,0,0}];

i23 : c_13=z_8^3;

i24 : c_15=-z_8^3;

i25 : T=S[x_0..x_6];

i26 : I=ideal({
      x_0^2+x_1^2 + x_2^2,
      x_1*x_2+x_3*x_6 + i*x_4*x_5,
      x_3*x_4+x_5*x_6,
      x_1^2 - x_2^2+c_8*(x_3*x_6 - i*x_4*x_5),
      x_0*x_1+c_10*(x_3^2 + x_4^2)+c_11*(x_5^2 -x_6^2),
      x_0*x_2+c_10*(i*x_3^2 - i*x_4^2)+c_11*(i*x_5^2 + i*x_6^2),
      x_0*x_5+c_13*(i*x_1*x_4 - x_2*x_4),
      x_0*x_6+c_13*(x_1*x_3 -i*x_2*x_3),
      x_0*x_3+c_15*(i*x_1*x_6 - x_2*x_6),
      x_0*x_4+c_15*(x_1*x_5 -i*x_2*x_5)
      });

o26 : Ideal of T

i27 : L=flatten entries gens gb(I,DegreeLimit=>6);

i28 : L3=select(L, i -> degree i == {3,0});

i29 : L3c=unique apply(L3, i -> leadCoefficient i);

i30 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
c_8*c_10-ww_8*c_10^2+ww_8^2*c_8*c_11-ww_8*c_11^2+2*ww_8*c_8-2*ww_8^2*c_10-2*c_11
c_8-2*ww_8*c_10-2*ww_8^2
c_8+2*ww_8^3*c_11+2*ww_8^2
c_10+ww_8^2*c_11+2*ww_8
c_10
1

This suggests setting \( c_8- 2 \zeta_8 c_{10}-2 i=0\) and \(c_8+2 \zeta_8^3 c_{11}+2 i = 0\).

This leads to the following conjectural description of the desired family: for generic values of \(c_8\), \[ \begin{array}{l} x_0^2+x_1^2 + x_2^2,\\ x_1 x_2+x_3 x_6 + i x_4 x_5,\\ x_3 x_4+x_5 x_6,\\ x_1^2 - x_2^2+c_8 (x_3 x_6 - i x_4 x_5),\\ x_0 x_1+c_{10} (x_3^2 + x_4^2)+c_{11} (x_5^2 -x_6^2),\\ x_0 x_2+c_{10} (i x_3^2 - i x_4^2)+c_{11} (i x_5^2 + i x_6^2),\\ x_0 x_5+\zeta_8^3 (i x_1 x_4 - x_2 x_4),\\ x_0 x_6+\zeta_8^3 (x_1 x_3 -i x_2 x_3),\\ x_0 x_3-\zeta_8^3 (i x_1 x_6 - x_2 x_6),\\ x_0 x_4-\zeta_8^3 (x_1 x_5 -i x_2 x_5) \end{array} \] where \[ c_{10} = \frac{1}{2} \zeta_8^{-1} c_8 -\zeta_8 \] and \[ c_{11} = \frac{1}{2} \zeta_8 c_8 -\zeta_8^{-1}. \] In the next section we show that at least two different values of \(c_{8}\) yield equations of a smooth curve with the correct automorphisms.

Checking the equations in `Magma`

We check that for two different values of \(c_{8}\), we obtain a a smooth curve with the correct automorphisms. This implies that a general member of the family is a smooth curve with the correct automorphisms. However, I do not know how to show that the two curves studied below are not isomorphic to each other; it is possible that we have described a point in \( \mathcal{M}_{g}\) rather than a curve in \( \mathcal{M}_g\).

First we check the value \(c_{8}=1\):

Magma V2.21-10    Wed Mar 16 2016 18:03:45 on ace-math01 [Seed = 413636336]
Type ? for help.  Type -D to quit.
> K<z_8>:=CyclotomicField(8);
> i:=z_8^2;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_8:=1;
> c_10:=(1/2)*z_8^(-1)*c_8 -z_8; 
> c_11:=(1/2)*z_8*c_8 -z_8^(-1);
> X:=Scheme(P6,[
> x_0^2+x_1^2 + x_2^2,
> x_1*x_2+x_3*x_6 + i*x_4*x_5,
> x_3*x_4+x_5*x_6,
> x_1^2 - x_2^2+c_8*(x_3*x_6 - i*x_4*x_5),
> x_0*x_1+c_10*(x_3^2 + x_4^2)+c_11*(x_5^2 -x_6^2),
> x_0*x_2+c_10*(i*x_3^2 - i*x_4^2)+c_11*(i*x_5^2 + i*x_6^2),
> x_0*x_5+z_8^3*(i*x_1*x_4 - x_2*x_4),
> x_0*x_6+z_8^3*(x_1*x_3 -i*x_2*x_3),
> x_0*x_3-z_8^3*(i*x_1*x_6 - x_2*x_6),
> x_0*x_4-z_8^3*(x_1*x_5 -i*x_2*x_5)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> GL7K:=GeneralLinearGroup(7,K);
> A:=Matrix([
> [-1,0,0,0,0,0,0],
> [0,0,-1,0,0,0,0],
> [0,-1,0,0,0,0,0],
> [0,0,0,0,z_8,0,0],
> [0,0,0,-z_8^3,0,0,0],
> [0,0,0,0,0,0,z_8^3],
> [0,0,0,0,0,-z_8,0]
> ]);
> B:=Matrix([
> [-1,0,0,0,0,0,0],
> [0,1,0,0,0,0,0],
> [0,0,-1,0,0,0,0],
> [0,0,0,0,-z_8^2,0,0],
> [0,0,0,z_8^2,0,0,0],
> [0,0,0,0,0,0,-1],
> [0,0,0,0,0,-1,0]
> ]);
> C:=Matrix([
> [-1,0,0,0,0,0,0],
> [0,-1,0,0,0,0,0],
> [0,0,-1,0,0,0,0],
> [0,0,0,-1,0,0,0],
> [0,0,0,0,1,0,0],
> [0,0,0,0,0,1,0],
> [0,0,0,0,0,0,-1]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
2
> IdentifyGroup(sub<GL7K | A,B,C>);
<32, 42>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_2
-x_1
-z_8^3*x_4
z_8*x_3
-z_8*x_6
z_8^3*x_5
and inverse
-x_0
-x_2
-x_1
-z_8^3*x_4
z_8*x_3
-z_8*x_6
z_8^3*x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1
-x_2
z_8^2*x_4
-z_8^2*x_3
-x_6
-x_5
and inverse
-x_0
x_1
-x_2
z_8^2*x_4
-z_8^2*x_3
-x_6
-x_5
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
-x_3
x_4
x_5
-x_6
and inverse
-x_0
-x_1
-x_2
-x_3
x_4
x_5
-x_6

Next, we check the value \(c_{8}=13-\zeta_{24}^5\):

> K<z_24>:=CyclotomicField(24);
> z_8:=z_24^3;
> i:=z_8^2;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_8:=13-z_24^5;
> c_10:=(1/2)*z_8^(-1)*c_8 -z_8; 
> c_11:=(1/2)*z_8*c_8 -z_8^(-1);
> X:=Scheme(P6,[
> x_0^2+x_1^2 + x_2^2,
> x_1*x_2+x_3*x_6 + i*x_4*x_5,
> x_3*x_4+x_5*x_6,
> x_1^2 - x_2^2+c_8*(x_3*x_6 - i*x_4*x_5),
> x_0*x_1+c_10*(x_3^2 + x_4^2)+c_11*(x_5^2 -x_6^2),
> x_0*x_2+c_10*(i*x_3^2 - i*x_4^2)+c_11*(i*x_5^2 + i*x_6^2),
> x_0*x_5+z_8^3*(i*x_1*x_4 - x_2*x_4),
> x_0*x_6+z_8^3*(x_1*x_3 -i*x_2*x_3),
> x_0*x_3-z_8^3*(i*x_1*x_6 - x_2*x_6),
> x_0*x_4-z_8^3*(x_1*x_5 -i*x_2*x_5)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> A:=Matrix([
> [-1,0,0,0,0,0,0],
> [0,0,-1,0,0,0,0],
> [0,-1,0,0,0,0,0],
> [0,0,0,0,z_8,0,0],
> [0,0,0,-z_8^3,0,0,0],
> [0,0,0,0,0,0,z_8^3],
> [0,0,0,0,0,-z_8,0]
> ]);
> B:=Matrix([
> [-1,0,0,0,0,0,0],
> [0,1,0,0,0,0,0],
> [0,0,-1,0,0,0,0],
> [0,0,0,0,-z_8^2,0,0],
> [0,0,0,z_8^2,0,0,0],
> [0,0,0,0,0,0,-1],
> [0,0,0,0,0,-1,0]
> ]);
> C:=Matrix([
> [-1,0,0,0,0,0,0],
> [0,-1,0,0,0,0,0],
> [0,0,-1,0,0,0,0],
> [0,0,0,-1,0,0,0],
> [0,0,0,0,1,0,0],
> [0,0,0,0,0,1,0],
> [0,0,0,0,0,0,-1]
> ]);
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_2
-x_1
(-z_24^5 + z_24)*x_4
z_24^3*x_3
-z_24^3*x_6
(z_24^5 - z_24)*x_5
and inverse
-x_0
-x_2
-x_1
(-z_24^5 + z_24)*x_4
z_24^3*x_3
-z_24^3*x_6
(z_24^5 - z_24)*x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1
-x_2
z_24^6*x_4
-z_24^6*x_3
-x_6
-x_5
and inverse
-x_0
x_1
-x_2
z_24^6*x_4
-z_24^6*x_3
-x_6
-x_5
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
-x_3
x_4
x_5
-x_6
and inverse
-x_0
-x_1
-x_2
-x_3
x_4
x_5
-x_6

Computing the Betti table in `Macaulay2`

We use Macaulay2 to compute the Betti table of the ideal generated by these equations.

Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(8);

i3 : z_8=K_0;

i4 : i=z_8^2;

i5 : R=K[x_0..x_6]

o5 = R

o5 : PolynomialRing

i6 : c_8=1;

i7 : c_10=(1/2)*z_8^(-1)*c_8 -z_8; 

i8 : c_11=(1/2)*z_8*c_8 -z_8^(-1);

i9 : I=ideal({
     x_0^2+x_1^2 + x_2^2,
     x_1*x_2+x_3*x_6 + i*x_4*x_5,
     x_3*x_4+x_5*x_6,
     x_1^2 - x_2^2+c_8*(x_3*x_6 - i*x_4*x_5),
     x_0*x_1+c_10*(x_3^2 + x_4^2)+c_11*(x_5^2 -x_6^2),
     x_0*x_2+c_10*(i*x_3^2 - i*x_4^2)+c_11*(i*x_5^2 + i*x_6^2),
     x_0*x_5+z_8^3*(i*x_1*x_4 - x_2*x_4),
     x_0*x_6+z_8^3*(x_1*x_3 -i*x_2*x_3),
     x_0*x_3-z_8^3*(i*x_1*x_6 - x_2*x_6),
     x_0*x_4-z_8^3*(x_1*x_5 -i*x_2*x_5)
     });

o9 : Ideal of R

i10 : betti res I

             0  1  2  3  4 5
o10 = total: 1 10 19 19 10 1
          0: 1  .  .  .  . .
          1: . 10 16  3  . .
          2: .  .  3 16 10 .
          3: .  .  .  .  . 1

o10 : BettiTally

By [Schreyer1986], this Betti table implies that this curve has a \(g_4^1\). We get the same Betti table with \(c_{8} = 13-\zeta_{24}^5\).

Finding equations of a 1-parameter family of genus 7 Riemann surfaces with 32 automorphisms

Obtaining candidate polynomials in Magma

Flattening stratification in Macaulay2

Checking the equations in Magma

Computing the Betti table in Macaulay2

Obtaining candidate polynomials in `Magma`

Flattening stratification in `Macaulay2`

Checking the equations in `Magma`

Computing the Betti table in `Macaulay2`