Fordham University

Finding equations of a genus 7 Riemann surface with 32 automorphisms

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 (32,10) 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 (4,4,8).

We use Macaulay2 and Magma to compute equations of this Riemann surface. The main tools are the Eichler trace formula, black-box commands in Magma for obtaining matrix generators of a representation of a finite group having a specified character, and a partial computation of a flattening stratification using Gröbner bases in Macaulay2

Obtaining candidate polynomials in Magma and GAP

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. 
Davids-MacBook-Pro-2:~ dswinarski$ ace1
Last login: Sun Nov  8 21:40:51 2015 from 100.44.116.13
For any questions, contact Paul Smith, psmith66@fordham.edu
See http://faculty.fordham.edu/rcc for news and help.
[dswinarski@ace-math01 ~]$ magma
Magma V2.21-4     Sun Nov  8 2015 21:50:47 on ace-math01 [Seed = 1902587270]
Type ? for help.  Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> G:=SmallGroup(32,10);
> G;
GrpPC : G of order 32 = 2^5
PC-Relations:
    G.1^2 = G.4, 
    G.2^2 = G.5, 
    G.3^2 = G.5, 
    G.2^G.1 = G.2 * G.3, 
    G.3^G.1 = G.3 * G.5, 
    G.3^G.2 = G.3 * G.5
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,7,[4,4,8]);
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  1  1  2  2  4  4  4  4   2   2   2   2
Order |   1  2  2  2  4  4  4  4  4  4   8   8   8   8
------------------------------------------------------
p  =  2   1  1  1  1  4  4  4  3  4  3   6   6   6   6
------------------------------------------------------
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   0   1 -1 -1  1  1 -1  1 -I -1  I  -I  -I   I   I
X.6   0   1 -1 -1  1  1 -1 -1 -I  1  I   I   I  -I  -I
X.7   0   1 -1 -1  1  1 -1 -1  I  1 -I  -I  -I   I   I
X.8   0   1 -1 -1  1  1 -1  1  I -1 -I   I   I  -I  -I
X.9   +   2  2  2  2 -2 -2  0  0  0  0   0   0   0   0
X.10  +   2 -2 -2  2 -2  2  0  0  0  0   0   0   0   0
X.11  -   2  2 -2 -2  0  0  0  0  0  0  Z1 -Z1 -Z1  Z1
X.12  0   2 -2  2 -2  0  0  0  0  0  0  Z2 -Z2  Z2 -Z2
X.13  -   2  2 -2 -2  0  0  0  0  0  0 -Z1  Z1  Z1 -Z1
X.14  0   2 -2  2 -2  0  0  0  0  0  0 -Z2  Z2 -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.4 * G.5

[3]     Order 2       Length 1      
        Rep G.4

[4]     Order 2       Length 1      
        Rep G.5

[5]     Order 4       Length 2      
        Rep G.3

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

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

[8]     Order 4       Length 4      
        Rep G.1 * G.4

[9]     Order 4       Length 4      
        Rep G.2

[10]    Order 4       Length 4      
        Rep G.1

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

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

[13]    Order 8       Length 2      
        Rep G.1 * G.2 * G.4

[14]    Order 8       Length 2      
        Rep G.1 * G.2 * G.4 * G.5


Surface kernel generators:  [ G.2 * G.3, G.1 * 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, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0 ]
I_2      =[ 0, 1, 1, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0 ]
S_2      =[ 0, 2, 2, 2, 1, 1, 1, 1, 4, 2, 0, 1, 1, 1 ]
H^0(C,2K)=[ 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1 ]
I_3      =[ 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 7, 4, 6, 5 ]
S_3      =[ 1, 1, 1, 1, 0, 2, 2, 2, 2, 4, 9, 6, 9, 7 ]
H^0(C,3K)=[ 1, 1, 0, 1, 0, 1, 1, 1, 1, 2, 2, 2, 3, 2 ]
I2timesS1=[ 0, 0, 1, 0, 0, 1, 1, 1, 1, 2, 9, 6, 8, 7 ]
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
[
    [ z^8    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^8    0    0]
    [   0    0    0 -z^8    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   z^8     0     0     0     0]
    [    0   z^8     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]
]
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],

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

    [  z^8     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^4     0     0     0]
    [    0     0     0     0  z^12     0     0]
    [    0     0     0     0     0  z^12     0]
    [    0     0     0     0     0     0  -z^4]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 2
Dimension 2
Multiplicity 2
[
    x_1*x_6 + z^12*x_2*x_5,
    x_3*x_4
]
I2 contains a 1-dimensional subspace of CharacterRow 3
Dimension 2
Multiplicity 2
[
    x_1*x_2,
    x_5*x_6
]
I2 contains a 1-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 2
[
    x_0^2,
    x_1*x_6 - z^12*x_2*x_5
]
I2 contains a 1-dimensional subspace of CharacterRow 6
Dimension 1
Multiplicity 1
[
    x_3*x_6 - z^8*x_4*x_5
]
I2 contains a 4-dimensional subspace of CharacterRow 9
Dimension 8
Multiplicity 4
[
    x_1^2,
    x_1*x_5,
    x_2^2,
    x_2*x_6,
    x_3^2,
    x_4^2,
    x_5^2,
    x_6^2
]
I2 contains a 2-dimensional subspace of CharacterRow 10
Dimension 4
Multiplicity 2
[
    x_1*x_3,
    x_2*x_4,
    x_3*x_5,
    x_4*x_6
]
> GL7K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL7K | MatrixGens>;
> FindParallelBases(MatrixG,[Q[5][1],Q[5][3]],[Q[5][2],Q[5][4]]);
[-z^4*x_2*x_6]
[     x_1*x_5]
> FindParallelBases(MatrixG,[Q[5][1],Q[5][3]],[Q[5][5],Q[5][6]]);
[-z^8*x_4^2]
[     x_3^2]
> FindParallelBases(MatrixG,[Q[5][1],Q[5][3]],[Q[5][7],Q[5][8]]);
[    x_5^2]
[z^8*x_6^2]
> FindParallelBases(MatrixG,[Q[6][1],Q[6][2]],[Q[6][3],Q[6][4]]);
[z^12*x_4*x_6]
[     x_3*x_5]

The output above shows that the ideal contains quadrics from three isotypical subspaces of \(S_2\). Note that the power of z\(=\zeta_{32}\) in our equations is always a multiple of 4. Therefore in the sequel we reduce these to z_8\(= \zeta_{8}\).

The first isotypical subspace, which corresponds to the character \( \chi_2\) in the character table shown above, yields a polynomial of the form \[ c_1(x_1 x_6 + z_8^{3} x_2 x_5) + c_2(x_3 x_4) \] Assume that \(c_1\) and \(c_2\) are nonzero. Then by scaling \(x_3\), we may assume that \( c_1 = c_2 = 1\).

The second isotypical subspace, which corresponds to the character \( \chi_3\) in the character table shown above, yields a polynomial of the form \[ c_3(x_1 x_2 ) + c_4(x_5 x_6) \] Assume that \( c_3 \) and \(c_4\) are nonzero. Then by scaling \( x_5\), we may assume that \( c_3 = c_4 =1\).

The third isotypical subspace, which corresponds to the character \( \chi_4\) in the character table shown above, yields a polynomial of the form \[ c_5(x_0^2) + c_6(x_1 x_6 - \zeta_{8}^3 x_2 x_5) \] Assume that \( c_5 \) and \(c_6\) are nonzero. Then by scaling \( x_0\), we may assume that \( c_5 = c_6 =1\).

The fourth isotypical subspace, which corresponds to the character \( \chi_6\) in the character table shown above, yields the polynomial \[ x_3 x_6 - \zeta_{8}^2 x_4 x_5. \]

The fifth isotypical subspace corresponds to the character \( \chi_{9}\) 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 \( \operatorname{Span}\langle x_1^2, x_2^2 \rangle \) generate one copy of \(V_{9}\) and use the FindParallelBases function to find ordered bases of \( \operatorname{Span}\langle x_1 x_5, x_2 x_6\rangle \), \( \operatorname{Span}\langle x_3^2, x_4^2\rangle \), and \( \operatorname{Span}\langle x_5^2, x_6^2\rangle \), such that the action of \(G\) is given by the same matrices relative to all four ordered bases. This yields the bases 

[-z^4*x_2*x_6]
[     x_1*x_5],
[-z^8*x_4^2]
[     x_3^2],
[    x_5^2]
[z^8*x_6^2]
Therefore the candidate polynomials for this isotypical subspace are \[ \begin{array}{l} c_7 (x_1^2)+c_8 (-\zeta_8 x_2 x_6)+c_9 (-\zeta_8^2 x_4^2)+c_{10} (x_5^2),\\ c_7 (x_2^2)+c_8 (x_1 x_5)+c_9(x_3^2)+c_{10} (\zeta_8^2 x_6^2),\\ c_{11} (x_1^2)+ c_{12} (-\zeta_8 x_2 x_6)+c_{13} (-\zeta_8^2 x_4^2)+c_{14} (x_5^2),\\ c_{11} (x_2^2)+c_{12} (x_1 x_5)+c_{13} (x_3^2)+c_{14} (\zeta_8^2 x_6^2), \end{array} \] Assume that \( (c_7,c_8) \) and \( (c_{11},c_{12}) \) are linearly independent. Then after row reducing we may assume that \( c_7 = 1\), \(c_8 = 0\), \(c_{11} = 0\), and \(c_{12}=1\).

The sixth isotypical subspace corresponds to the character \( \chi_{10}\) in the character table shown above. From the block decomposition of the matrix surface kernal generators, we let \( \operatorname{Span}\langle x_1 x_3, x_2 x_4 \rangle \) generate one copy of \(V_{10}\) and use the FindParallelBases function to find an ordered bases of \( \operatorname{Span}\langle x_3 x_5, x_4 x_6\rangle \) such that the action of \(G\) is given by the same matrices relative to both ordered bases. This yields the basis 


[z^12*x_4*x_6]
[     x_3*x_5]
which in turn suggests the polynomials \[ \begin{array}{l} c_{15} (x_1 x_3)+c_{16} (z^12 x_4 x_6),\\ c_{15} (x_2 x_4)+c_{16} (x_3 x_5) \end{array} \]

Assume that \(c_{15} \) and \(c_{16}\) are nonzero. Then after dividing by \(c_{15}\) we may assume that \(c_{15} = 1\). Thus we obtain the following candidate polynomials and fixed coefficients: \[ \begin{array}{l} c_1=c_2 = c_3 = c_4 =c_5 = c_6 = c_7 = c_{12} = c_{15} = 1;\\ c_8 = c_{11} = 0;\\ c_{1} (x_1 x_6 + \zeta_{8}^3 x_2 x_5)+c_{2} (x_3 x_4),\\ c_{3} (x_1 x_2)+c_{4} (x_5 x_6),\\ c_{5} (x_0^2)+c_{6} (x_1 x_6 - \zeta_{8}^3 x_2 x_5),\\ x_3 x_6 - \zeta_{8}^2 x_4 x_5,\\ c_{7} (x_1^2)+c_{8} (-\zeta_{8} x_2 x_6)+c_{9} (-\zeta_{8}^2 x_4^2)+c_{10} (x_5^2),\\ c_{7} (x_2^2)+c_{8} (x_1 x_5)+c_{9} (x_3^2)+c_{10} (\zeta_{8}^2 x_6^2),\\ c_{11} (x_1^2)+c_{12} (-\zeta_{8}^4 x_2 x_6)+c_{13} (-\zeta_{8}^2 x_4^2)+c_{14} (x_5^2),\\ c_{11} (x_2^2)+c_{12} (x_1 x_5)+c_{13} (x_3^2)+c_{14} (\zeta_{8}^2 x_6^2),\\ c_{15} (x_1 x_3)+c_{16} (\zeta_{8}^3 x_4 x_6),\\ c_{15} (x_2 x_4)+c_{16} (x_3 x_5)\\ \end{array} \] For generic values of \(c_9,c_{10},c_{13},c_{14},c_{16}\), the intersection of these 10 quadrics in \(\mathbb{P}^6\) is empty. Here is an example showing this: 

> K<z>:=CyclotomicField(32);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c1:=1;
> c2:=1;
> c3:=1;
> c4:=1;
> c5:=1;
> c6:=1;
> c7:=1;
> c8:=0;
> c9:=1;
> c10:=1;
> c11:=0;
> c12:=1;
> c13:=1;
> c14:=1;
> c15:=1;
> c16:=1;
> X:=Scheme(P6,[
> c1*(x_1*x_6 + z^12*x_2*x_5)+c2*(x_3*x_4),
> c3*(x_1*x_2)+c4*(x_5*x_6),
> c5*(x_0^2)+c6*(x_1*x_6 - z^12*x_2*x_5),
> x_3*x_6 - z^8*x_4*x_5,
> c7*(x_1^2)+c8*(-z^4*x_2*x_6)+c9*(-z^8*x_4^2)+c10*(x_5^2),
> c7*(x_2^2)+c8*(x_1*x_5)+c9*(x_3^2)+c10*(z^8*x_6^2),
> c11*(x_1^2)+c12*(-z^4*x_2*x_6)+c13*(-z^8*x_4^2)+c14*(x_5^2),
> c11*(x_2^2)+c12*(x_1*x_5)+c13*(x_3^2)+c14*(z^8*x_6^2),
> c15*(x_1*x_3)+c16*(z^12*x_4*x_6),
> c15*(x_2*x_4)+c16*(x_3*x_5)
> ]);
> Dimension(X);
-1
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(32);

i3 : z=K_0;

i4 : c1=1;

i5 : c2=1;

i6 : c3=1;

i7 : c4=1;

i8 : c5=1;

i9 : c6=1;

i10 : c7=1;

i11 : c8=0;

i12 : c11=0;

i13 : c12=1;

i14 : c15=1;

i15 : S=K[c9,c10,c13,c14,c16,Degrees=>{0,0,0,0,0}]

o15 = S

o15 : PolynomialRing

i16 : T=S[x_0..x_6];

i17 : I=ideal({
      c1*(x_1*x_6 + z^12*x_2*x_5)+c2*(x_3*x_4),
      c3*(x_1*x_2)+c4*(x_5*x_6),
      c5*(x_0^2)+c6*(x_1*x_6 - z^12*x_2*x_5),
      x_3*x_6 - z^8*x_4*x_5,
      c7*(x_1^2)+c8*(-z^4*x_2*x_6)+c9*(-z^8*x_4^2)+c10*(x_5^2),
      c7*(x_2^2)+c8*(x_1*x_5)+c9*(x_3^2)+c10*(z^8*x_6^2),
      c11*(x_1^2)+c12*(-z^4*x_2*x_6)+c13*(-z^8*x_4^2)+c14*(x_5^2),
      c11*(x_2^2)+c12*(x_1*x_5)+c13*(x_3^2)+c14*(z^8*x_6^2),
      c15*(x_1*x_3)+c16*(z^12*x_4*x_6),
      c15*(x_2*x_4)+c16*(x_3*x_5)
      });

o17 : Ideal of T

i18 : 
      L=flatten entries gens gb(I,DegreeLimit=>4);

i19 : tally apply(L, i -> degree i)

o19 = Tally{{2, 0} => 10}
            {3, 0} => 9

o19 : Tally

i20 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
(c9-ww_32^4*c16)*x_4^2*x_6+(ww_32^8*c10-ww_32^4)*x_5^2*x_6
c14*x_5^3+(ww_32^12*c13-ww_32^4)*x_2*x_5*x_6+c13*x_1*x_6^2
c16*x_3*x_5^2-ww_32^8*x_2*x_3*x_6
c14*x_3*x_5^2+(ww_32^12*c13-ww_32^4)*x_2*x_3*x_6+ww_32^8*c13*x_1*x_4*x_6
c10*x_3*x_5^2+ww_32^12*c9*x_2*x_3*x_6+(ww_32^8*c9-ww_32^12*c16)*x_1*x_4*x_6
x_2*x_5^2+ww_32^12*x_3^2*x_6-ww_32^4*x_1*x_5*x_6
(c9-ww_32^4*c16)*x_3^2*x_5+(ww_32^8*c10-ww_32^4)*x_5*x_6^2
c13*x_2*x_3*x_5+(-c13*c16+ww_32^12*c14-ww_32^8*c16)*x_4*x_6^2
(c9-ww_32^4*c16)*x_2*x_3*x_5+(-c9*c16+ww_32^12*c10)*x_4*x_6^2

The leading coefficients of the generators above suggest the choice \( c_9 = z_{32}^4 c_{16}\). We rerun the calculation with this choice. 


i21 : 
      loadPackage("Cyclotomic");

i22 : K=cyclotomicField(32);

i23 : z=K_0;

i24 : c1=1;

i25 : c2=1;

i26 : c3=1;

i27 : c4=1;

i28 : c5=1;

i29 : c6=1;

i30 : c7=1;

i31 : c8=0;

i32 : c11=0;

i33 : c12=1;

i34 : c15=1;

i35 : S=K[c10,c13,c14,c16,Degrees=>{0,0,0,0}]

o35 = S

o35 : PolynomialRing

i36 : c9=z^4*c16;

i37 : T=S[x_0..x_6];

i38 : I=ideal({
      c1*(x_1*x_6 + z^12*x_2*x_5)+c2*(x_3*x_4),
      c3*(x_1*x_2)+c4*(x_5*x_6),
      c5*(x_0^2)+c6*(x_1*x_6 - z^12*x_2*x_5),
      x_3*x_6 - z^8*x_4*x_5,
      c7*(x_1^2)+c8*(-z^4*x_2*x_6)+c9*(-z^8*x_4^2)+c10*(x_5^2),
      c7*(x_2^2)+c8*(x_1*x_5)+c9*(x_3^2)+c10*(z^8*x_6^2),
      c11*(x_1^2)+c12*(-z^4*x_2*x_6)+c13*(-z^8*x_4^2)+c14*(x_5^2),
      c11*(x_2^2)+c12*(x_1*x_5)+c13*(x_3^2)+c14*(z^8*x_6^2),
      c15*(x_1*x_3)+c16*(z^12*x_4*x_6),
      c15*(x_2*x_4)+c16*(x_3*x_5)
      });

o38 : Ideal of T

i39 : 
      L=flatten entries gens gb(I,DegreeLimit=>4);

i40 : tally apply(L, i -> degree i)

o40 = Tally{{2, 0} => 10}
            {3, 0} => 8

o40 : Tally

i41 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
(c10+ww_32^12)*x_5*x_6^2
(c10+ww_32^12)*x_5^2*x_6
c14*x_5^3+(ww_32^12*c13-ww_32^4)*x_2*x_5*x_6+c13*x_1*x_6^2
c16*x_3*x_5^2-ww_32^8*x_2*x_3*x_6
c14*x_3*x_5^2+(ww_32^12*c13-ww_32^4)*x_2*x_3*x_6+ww_32^8*c13*x_1*x_4*x_6
c10*x_3*x_5^2-c16*x_2*x_3*x_6
x_2*x_5^2+ww_32^12*x_3^2*x_6-ww_32^4*x_1*x_5*x_6
c13*x_2*x_3*x_5+(-c13*c16+ww_32^12*c14-ww_32^8*c16)*x_4*x_6^2

The leading coefficients of the generators above suggest the choice \( c_{10}= -\zeta_{32}^{12}\). We rerun the calculation with this choice. 


i42 : c1=1;

i43 : c2=1;

i44 : c3=1;

i45 : c4=1;

i46 : c5=1;

i47 : c6=1;

i48 : c7=1;

i49 : c8=0;

i50 : c11=0;

i51 : c12=1;

i52 : c15=1;

i53 : S=K[c13,c14,c16,Degrees=>{0,0,0}]

o53 = S

o53 : PolynomialRing

i54 : c9=z^4*c16;

i55 : c10=-z^12;

i56 : T=S[x_0..x_6];

i57 : I=ideal({
      c1*(x_1*x_6 + z^12*x_2*x_5)+c2*(x_3*x_4),
      c3*(x_1*x_2)+c4*(x_5*x_6),
      c5*(x_0^2)+c6*(x_1*x_6 - z^12*x_2*x_5),
      x_3*x_6 - z^8*x_4*x_5,
      c7*(x_1^2)+c8*(-z^4*x_2*x_6)+c9*(-z^8*x_4^2)+c10*(x_5^2),
      c7*(x_2^2)+c8*(x_1*x_5)+c9*(x_3^2)+c10*(z^8*x_6^2),
      c11*(x_1^2)+c12*(-z^4*x_2*x_6)+c13*(-z^8*x_4^2)+c14*(x_5^2),
      c11*(x_2^2)+c12*(x_1*x_5)+c13*(x_3^2)+c14*(z^8*x_6^2),
      c15*(x_1*x_3)+c16*(z^12*x_4*x_6),
      c15*(x_2*x_4)+c16*(x_3*x_5)
      });

o57 : Ideal of T

i58 : 
      
      L=flatten entries gens gb(I);

i59 : tally apply(L, i -> degree i)

o59 = Tally{{2, 0} => 10}
            {3, 0} => 20
            {4, 0} => 23
            {5, 0} => 1

o59 : Tally

i60 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
(c14^2*c16^2-ww_32^4*c14^2+ww_32^12*c16^2+1)*x_6^3
(c16^2-ww_32^4)*x_4*x_6^2
(c16^2-ww_32^4)*x_3*x_6^2
(c14^2*c16^2-ww_32^4*c14^2+ww_32^12*c16^2+1)*x_1*x_6^2
(c16^2-ww_32^4)*x_3*x_5*x_6
(c16^2-ww_32^4)*x_2*x_5*x_6+(-ww_32^4*c16^2+ww_32^8)*x_1*x_6^2
(c13*c14+ww_32^12*c13*c16+ww_32^8*c14-ww_32^4*c16)*x_2*x_5*x_6+(ww_32^12*c14^2*c16-ww_32^4*c13*c14+c13*c16-ww_32^12*c14)*x_1*x_6^2
(c16^2-ww_32^4)*x_1*x_5*x_6+(ww_32^8*c14*c16^2-ww_32^12*c14)*x_6^3
(c13*c14+ww_32^12*c13*c16+ww_32^8*c14-ww_32^4*c16)*x_1*x_5*x_6+(-ww_32^4*c13*c14*c16-c14^2-ww_32^12*c14*c16+ww_32^4*c13)*x_6^3
(c13*c16^2-ww_32^4*c13)*x_1*x_4*x_6
(c14+ww_32^12*c16)*x_3^2*x_6+(ww_32^8*c14-ww_32^4*c16)*x_1*x_5*x_6+(-ww_32^12*c14*c16+ww_32^12)*x_6^3
(c16^2-ww_32^4)*x_3^2*x_6
(c16^2-ww_32^4)*x_2*x_3*x_6
(c14*c16+ww_32^8*c13-1)*x_2*x_3*x_6+ww_32^4*c13*x_1*x_4*x_6
(c13*c16-ww_32^12*c14+ww_32^8*c16)*x_2*x_3*x_6-ww_32^12*c13*c16*x_1*x_4*x_6
(c13^2+ww_32^4*c14^2+2*ww_32^8*c13-1)*x_2*x_3*x_6+(ww_32^4*c13*c14*c16-ww_32^12*c13^2+ww_32^4*c13)*x_1*x_4*x_6
x_5^3+(ww_32^12*c13*c16-ww_32^4*c16)*x_2*x_5*x_6+(c13*c16-ww_32^12*c14+ww_32^8*c16)*x_1*x_6^2
x_3*x_5^2-ww_32^4*c16*x_2*x_3*x_6
x_2*x_5^2+ww_32^12*x_3^2*x_6-ww_32^4*x_1*x_5*x_6
c13*x_2*x_3*x_5+(-c13*c16+ww_32^12*c14-ww_32^8*c16)*x_4*x_6^2
The leading coefficients of the generators above suggest the choice \( c_{16}= \zeta_{32}^2\). We rerun the calculation with this choice. 

i84 : c1=1;

i85 : c2=1;

i86 : c3=1;

i87 : c4=1;

i88 : c5=1;

i89 : c6=1;

i90 : c7=1;

i91 : c8=0;

i92 : c11=0;

i93 : c12=1;

i94 : c15=1;

i95 : S=K[c13,c14,Degrees=>{0,0}]

o95 = S

o95 : PolynomialRing

i96 : c16=z^2;

i97 : c9=z^4*c16;

i98 : c10=-z^12;

i99 : T=S[x_0..x_6];

i100 : I=ideal({
       c1*(x_1*x_6 + z^12*x_2*x_5)+c2*(x_3*x_4),
       c3*(x_1*x_2)+c4*(x_5*x_6),
       c5*(x_0^2)+c6*(x_1*x_6 - z^12*x_2*x_5),
       x_3*x_6 - z^8*x_4*x_5,
       c7*(x_1^2)+c8*(-z^4*x_2*x_6)+c9*(-z^8*x_4^2)+c10*(x_5^2),
       c7*(x_2^2)+c8*(x_1*x_5)+c9*(x_3^2)+c10*(z^8*x_6^2),
       c11*(x_1^2)+c12*(-z^4*x_2*x_6)+c13*(-z^8*x_4^2)+c14*(x_5^2),
       c11*(x_2^2)+c12*(x_1*x_5)+c13*(x_3^2)+c14*(z^8*x_6^2),
       c15*(x_1*x_3)+c16*(z^12*x_4*x_6),
       c15*(x_2*x_4)+c16*(x_3*x_5)
       });

o100 : Ideal of T

i101 : 
       L=flatten entries gens gb(I);

i102 : tally apply(L, i -> degree i)

o102 = Tally{{2, 0} => 10}
             {3, 0} => 8
             {4, 0} => 11
             {5, 0} => 1

o102 : Tally

i103 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
(c13*c14+ww_32^14*c13+ww_32^8*c14-ww_32^6)*x_2*x_5*x_6+(-ww_32^4*c13*c14+ww_32^14*c14^2+ww_32^2*c13-ww_32^12*c14)*x_1*x_6^2
(c13*c14+ww_32^14*c13+ww_32^8*c14-ww_32^6)*x_1*x_5*x_6+(-ww_32^6*c13*c14-c14^2+ww_32^4*c13-ww_32^14*c14)*x_6^3
(c14+ww_32^14)*x_3^2*x_6+(ww_32^8*c14-ww_32^6)*x_1*x_5*x_6+(-ww_32^14*c14+ww_32^12)*x_6^3
(c13-ww_32^10*c14+ww_32^8)*x_2*x_3*x_6-ww_32^12*c13*x_1*x_4*x_6
x_5^3+(ww_32^14*c13-ww_32^6)*x_2*x_5*x_6+(ww_32^2*c13-ww_32^12*c14+ww_32^10)*x_1*x_6^2
x_3*x_5^2-ww_32^6*x_2*x_3*x_6
x_2*x_5^2+ww_32^12*x_3^2*x_6-ww_32^4*x_1*x_5*x_6
c13*x_2*x_3*x_5+(-ww_32^2*c13+ww_32^12*c14-ww_32^10)*x_4*x_6^2
The leading coefficients of the generators above suggest the choices \( c_{14}= -\zeta_{32}^{14}\) and \(c_{13} = -\zeta_{32}^8-\zeta_{32}^{24}\). In the next section we show that these choices yield equations of the desired curve.

Checking the equations in Magma

After substituting the values of the coefficients \(c_i\) shown above and simplifying the equations, we check that we obtain a smooth genus 7 curve with the desired automorphisms. 

    Magma V2.21-10    Fri Mar 11 2016 16:58:32 on ace-math01 [Seed = 1741819490]
Type ? for help.  Type -D to quit.
> K<z_16>:=CyclotomicField(16);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> x_1*x_6+z_16^6*x_2*x_5+x_3*x_4,
> x_1*x_2+x_5*x_6,
> x_0^2+x_1*x_6-z_16^6*x_2*x_5,
> x_3*x_6-z_16^4*x_4*x_5,
> x_1^2-z_16^7*x_4^2-z_16^6*x_5^2,
> x_2^2+z_16^3*x_3^2-z_16^10*x_6^2,
> -z_16^2*x_2*x_6+(z_16^16+z_16^8)*x_4^2-z_16^7*x_5^2,
> x_1*x_5+(-z_16^12-z_16^4)*x_3^2-z_16^11*x_6^2,
> x_1*x_3+z_16^7*x_4*x_6,
> x_2*x_4+z_16*x_3*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_16^2,0,0],
> [0,0,0,-z_16^6,0,0,0],
> [0,0,0,0,0,0,-z_16^6],
> [0,0,0,0,0,-z_16^2,0]
> ]);
> B:=Matrix([
> [z_16^4,0,0,0,0,0,0],
> [0,0,-z_16^6,0,0,0,0],
> [0,-z_16^2,0,0,0,0,0],
> [0,0,0,0,-z_16^4,0,0],
> [0,0,0,z_16^4,0,0,0],
> [0,0,0,0,0,0,-z_16^4],
> [0,0,0,0,0,-z_16^4,0]
> ]);
> IdentifyGroup(sub<GL7K | A,B>);
<32, 10>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_2
-x_1
-z_16^6*x_4
-z_16^2*x_3
-z_16^2*x_6
-z_16^6*x_5
and inverse
-x_0
-x_2
x_1
z_16^6*x_4
z_16^2*x_3
z_16^2*x_6
z_16^6*x_5
> Automorphism(X,B); 
Mapping from: Sch: X to Sch: X
with equations : 
z_16^4*x_0
-z_16^2*x_2
-z_16^6*x_1
z_16^4*x_4
-z_16^4*x_3
-z_16^4*x_6
-z_16^4*x_5
and inverse
-z_16^4*x_0
z_16^2*x_2
z_16^6*x_1
z_16^4*x_4
-z_16^4*x_3
z_16^4*x_6
z_16^4*x_5

Computing the Betti table in Macaulay2

We use Macaulay2 to compute the Betti table of the ideal generated by these equations. 
Macaulay2, version 1.8.2
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(16);

i3 : z_16=K_0;

i4 : R=K[x_0..x_6];

i5 : I=ideal{
     x_1*x_6+z_16^6*x_2*x_5+x_3*x_4,
     x_1*x_2+x_5*x_6,
     x_0^2+x_1*x_6-z_16^6*x_2*x_5,
     x_3*x_6-z_16^4*x_4*x_5,
     x_1^2-z_16^7*x_4^2-z_16^6*x_5^2,
     x_2^2+z_16^3*x_3^2-z_16^10*x_6^2,
     -z_16^2*x_2*x_6+(z_16^16+z_16^8)*x_4^2-z_16^7*x_5^2,
     x_1*x_5+(-z_16^12-z_16^4)*x_3^2-z_16^11*x_6^2,
     x_1*x_3+z_16^7*x_4*x_6,
     x_2*x_4+z_16*x_3*x_5
     };

o5 : Ideal of R

i6 : betti res I

            0  1  2  3  4 5
o6 = total: 1 10 25 25 10 1
         0: 1  .  .  .  . .
         1: . 10 16  9  . .
         2: .  .  9 16 10 .
         3: .  .  .  .  . 1

o6 : BettiTally
By [Schreyer1986], this Betti table implies that the curve has a \( g^{2}_{6} \).