Fordham University

Finding equations of a genus 7 Riemann surface with 48 automorphisms

Magaard, Shaska, Shpectorov, and Völklein give tables of smooth curves 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 (48,32) in the GAP library of small groups. The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (3,4,6).

We use Macaulay2 and Magma to compute equations of this curve. 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. 
Magma V2.21-4     Tue Sep  8 2015 10:06:37 on ace-math01 [Seed = 2602575160]
Type ? for help.  Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> G:=SmallGroup(48,32);
> G;
GrpPC : G of order 48 = 2^4 * 3
PC-Relations:
    G.1^2 = Id(G), 
    G.2^3 = Id(G), 
    G.3^2 = G.5, 
    G.4^2 = G.5, 
    G.5^2 = Id(G), 
    G.3^G.2 = G.4, 
    G.4^G.2 = G.3 * G.4, 
    G.4^G.3 = G.4 * G.5
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,7,[3,4,6]);
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   4   4  6  6   4   4   4   4   4   4
Order |   1  2  2  2   3   3  4  4   6   6   6   6   6   6
----------------------------------------------------------
p  =  2   1  1  1  1   6   5  3  3   5   6   5   5   6   6
p  =  3   1  2  3  4   1   1  7  8   3   3   2   4   2   4
----------------------------------------------------------
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   0   1  1  1  1-1-J   J  1  1   J-1-J   J   J-1-J-1-J
X.4   0   1  1  1  1   J-1-J  1  1-1-J   J-1-J-1-J   J   J
X.5   0   1 -1  1 -1-1-J   J  1 -1   J-1-J  -J  -J 1+J 1+J
X.6   0   1 -1  1 -1   J-1-J  1 -1-1-J   J 1+J 1+J  -J  -J
X.7   -   2  2 -2 -2  -1  -1  0  0   1   1  -1   1  -1   1
X.8   -   2 -2 -2  2  -1  -1  0  0   1   1   1  -1   1  -1
X.9   0   2 -2 -2  2 1+J  -J  0  0   J-1-J   J  -J-1-J 1+J
X.10  0   2  2 -2 -2  -J 1+J  0  0-1-J   J 1+J-1-J  -J   J
X.11  0   2  2 -2 -2 1+J  -J  0  0   J-1-J  -J   J 1+J-1-J
X.12  0   2 -2 -2  2  -J 1+J  0  0-1-J   J-1-J 1+J   J  -J
X.13  +   3 -3  3 -3   0   0 -1  1   0   0   0   0   0   0
X.14  +   3  3  3  3   0   0 -1 -1   0   0   0   0   0   0


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

J = RootOfUnity(3)


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

[2]     Order 2       Length 1      
        Rep G.1

[3]     Order 2       Length 1      
        Rep G.5

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

[5]     Order 3       Length 4      
        Rep G.2^2

[6]     Order 3       Length 4      
        Rep G.2

[7]     Order 4       Length 6      
        Rep G.3

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

[9]     Order 6       Length 4      
        Rep G.2 * G.5

[10]    Order 6       Length 4      
        Rep G.2^2 * G.5

[11]    Order 6       Length 4      
        Rep G.1 * G.2

[12]    Order 6       Length 4      
        Rep G.1 * G.2 * G.5

[13]    Order 6       Length 4      
        Rep G.1 * G.2^2

[14]    Order 6       Length 4      
        Rep G.1 * G.2^2 * G.5


Surface kernel generators:  [ G.2 * G.3 * G.4, G.1 * G.4 * G.5, G.1 * G.2^2 * 
G.3 * G.4 * G.5 ]
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, 1, 1, 0, 0, 0, 1, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0 ]
I_2      =[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2 ]
S_2      =[ 0, 1, 0, 2, 0, 1, 1, 0, 1, 0, 1, 0, 2, 4 ]
H^0(C,2K)=[ 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2 ]
I_3      =[ 0, 1, 0, 0, 0, 0, 2, 3, 5, 4, 3, 5, 2, 1 ]
S_3      =[ 1, 2, 1, 0, 0, 0, 3, 4, 7, 6, 4, 7, 4, 2 ]
H^0(C,3K)=[ 1, 1, 1, 0, 0, 0, 1, 1, 2, 2, 1, 2, 2, 1 ]
I2timesS1=[ 0, 1, 0, 0, 0, 0, 4, 5, 6, 5, 4, 6, 2, 1 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 48 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 -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],

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

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

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

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

    [     z^8        0        0        0        0        0        0]
    [       0     -z^8        0        0        0        0        0]
    [       0       -1  z^8 - 1        0        0        0        0]
    [       0        0        0 -z^8 + 1       -1        0        0]
    [       0        0        0        0      z^8        0        0]
    [       0        0        0        0        0       -1        0]
    [       0        0        0        0        0 -z^8 + 1      z^8]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 2
[
    x_0^2,
    x_3*x_5 + z^8*x_3*x_6 + (-z^8 + 1)*x_4*x_6
]
I2 contains a 3-dimensional subspace of CharacterRow 13
Dimension 6
Multiplicity 2
[
    x_1*x_3 - x_2*x_4,
    x_1*x_4,
    x_1*x_5 + (-z^8 + 1)*x_2*x_5,
    x_1*x_6 + z^8*x_2*x_5,
    x_2*x_3,
    x_2*x_6
]
I2 contains a 6-dimensional subspace of CharacterRow 14
Dimension 12
Multiplicity 4
[
    x_1^2,
    x_1*x_2,
    x_2^2,
    x_3^2,
    x_3*x_4,
    x_3*x_5 + (z^8 - 1)*x_4*x_6,
    x_3*x_6,
    x_4^2,
    x_4*x_5 + z^8*x_4*x_6,
    x_5^2,
    x_5*x_6,
    x_6^2
      ]
> GL7K:=Parent(MatrixGens[1]);
> MatrixG:=sub;
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2],Q[2][5]],[Q[2][3],Q[2][4],Q[2][6]\
]);
[x_1*x_5 + 1/3*(2*z^8 - 1)*x_1*x_6 + 1/3*(-2*z^8 + 1)*x_2*x_5 + 1/3*(2*z^8 - 
    1)*x_2*x_6]
[1/6*(-2*z^8 + 1)*x_1*x_5 + 1/6*(2*z^8 - 1)*x_1*x_6 - 1/2*x_2*x_5 + 1/2*x_2*x_6]
[1/2*(-2*z^8 + 1)*x_1*x_5 + 1/2*x_1*x_6 - 1/2*x_2*x_5 + 1/6*(-2*z^8 + 
    1)*x_2*x_6]
> FindParallelBases(MatrixG,[Q[3][1],Q[3][2],Q[3][3]],[Q[3][4],Q[3][5],Q[3][8]\
]);
[   x_4^2]
[-x_3*x_4]
[   x_3^2]
> FindParallelBases(MatrixG,[Q[3][1],Q[3][2],Q[3][3]],[Q[3][6],Q[3][7],Q[3][9]\
]);
[x_3*x_5 - x_3*x_6 + 1/3*(-2*z^8 + 1)*x_4*x_5 + 1/3*(2*z^8 - 1)*x_4*x_6]
[1/3*(-2*z^8 + 1)*x_3*x_5 + 1/3*(2*z^8 - 1)*x_3*x_6 - x_4*x_5 + 1/3*(-2*z^8 + 
    1)*x_4*x_6]
[-x_3*x_5 + 1/3*(-2*z^8 + 1)*x_3*x_6 + (2*z^8 - 1)*x_4*x_5 - x_4*x_6]
> FindParallelBases(MatrixG,[Q[3][1],Q[3][2],Q[3][3]],[Q[3][10],Q[3][11],Q[3][\
12]]);
[                 x_5^2 + (z^8 - 1)*x_5*x_6 - z^8*x_6^2]
[1/2*(-2*z^8 + 1)*x_5^2 + z^8*x_5*x_6 - 1/2*x_6^2]
[                   (z^8 + 1)*x_5*x_6 + (z^8 - 1)*x_6^2]

The output above shows that the ideal contains quadrics from three isotypical subspaces of \(S_2\). Note that the power of z\(=\zeta_{48}\) in our equations is always a multiple of 8. Therefore in the sequel we reduce these to z_6\(= \zeta_{6}\).

The first isotypical subspace, which corresponds to the character \( \chi_4\) in the character table shown above, yields a polynomial of the form \[ c_1(x_0^2) + c_2(x_3 x_5 + \zeta_6 x_3 x_6 + (-\zeta_6+ 1) x_4 x_6) \] Assume that \(c_1\) and \(c_2\) are nonzero. Then by scaling \(x_0\), we may assume that this polynomial is \(x_0^2 + x_3 x_5 + \zeta_6 x_3 x_6 + (-\zeta_6 + 1) x_4 x_6\).

The second isotypical subspace corresponds to the character \( \chi_{13}\) 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 \( \operatorname{Span}\langle x_1 x_3 - x_2 x_4, x_1 x_4, x_2 x_3 \rangle \) generate one copy of \(V_{13}\) and use the FindParallelBases function to find an ordered basis \( Q_1,Q_2,Q_3 \) of \( \operatorname{Span}\langle x_1 x_5 + (-\zeta_6 + 1) x_2 x_5, x_1 x_6 + \zeta_6 x_2 x_5, x_2 x_6\rangle \) such that the action of \(G\) is given by the same matrices relative to the two ordered bases. This yielded 

[x_1*x_5 + 1/3*(2*z^8 - 1)*x_1*x_6 + 1/3*(-2*z^8 + 1)*x_2*x_5 + 1/3*(2*z^8 - 
    1)*x_2*x_6]
[1/6*(-2*z^8 + 1)*x_1*x_5 + 1/6*(2*z^8 - 1)*x_1*x_6 - 1/2*x_2*x_5 + 1/2*x_2*x_6]
[1/2*(-2*z^8 + 1)*x_1*x_5 + 1/2*x_1*x_6 - 1/2*x_2*x_5 + 1/6*(-2*z^8 + 
    1)*x_2*x_6]
Therefore the candidate polynomials for this isotypical subspace are \[ \begin{array}{l} c_3 (x_1 x_3 - x_2 x_4) + c_4(x_1 x_5 + 1/3(2\zeta_6 - 1) x_1 x_6 + 1/3 (-2 \zeta_6+ 1) x_2 x_5 + 1/3 (2 \zeta_6 - 1) x_2 x_6) \\ c_3 (x_1 x_4) + c_4(1/6 (-2 \zeta_6 + 1)x_1 x_5 + 1/6 (2 \zeta_6 - 1) x_1 x_6 - 1/2 x_2 x_5 + 1/2 x_2 x_6) \\ c_3 (x_2 x_3) + c_4(1/2 (-2 \zeta_6 + 1) x_1 x_5 + 1/2 x_1 x_6 - 1/2 x_2 x_5 + 1/6 (-2 \zeta_6 + 1) x_2 x_6) \end{array} \] Assume that \(c_3\) and \(c_4\) are nonzero. Then by scaling \( x_3\) and \( x_4\) we may assume that \(c_3 = c_4 = 1\). Thus, we obtain \[ \begin{array}{l} x_1 x_3 - x_2 x_4 + x_1 x_5 + 1/3(2\zeta_6 - 1) x_1 x_6 + 1/3 (-2 \zeta_6 + 1) x_2 x_5 + 1/3 (2 \zeta_6 - 1) x_2 x_6 \\ x_1 x_4 + 1/6 (-2 \zeta_6 + 1)x_1 x_5 + 1/6 (2 \zeta_6 - 1) x_1 x_6 - 1/2 x_2 x_5 + 1/2 x_2 x_6 \\ x_2 x_3+ 1/2 (-2 \zeta_6 + 1) x_1 x_5 + 1/2 x_1 x_6 - 1/2 x_2 x_5 + 1/6 (-2 \zeta_6 + 1) x_2 x_6 \end{array} \]

The third isotypical subspace corresponds to the character \( \chi_{14}\) in the character table shown above. First, we use the FindParallelBases command to obtain candidate polynomials of the following form: \[ \begin{array}{l} c_5 x_1^2+c_6 (x_4^2)+c_7 (x_3 x_5 - x_3 x_6 + 1/3 (-2 \zeta_6 + 1) x_4 x_5 + 1/3 (2 \zeta_6 - 1) x_4 x_6)+c_8 (x_5^2 + (\zeta_6 - 1) x_5 x_6 -\zeta_6 x_6^2) \\ c_5 x_1 x_2+c_6 (-x_3 x_4)+c_7 (1/3 (-2 \zeta_6 + 1) x_3 x_5 + 1/3 (2 \zeta_6 - 1) x_3 x_6 - x_4 x_5 + 1/3 (-2 \zeta_6+ 1) x_4 x_6)+c_8 (1/2 (-2 \zeta_6 + 1) x_5^2 + \zeta_6 x_5 x_6 - 1/2 x_6^2) \\ c_5 x_2^2+c_6 (x_3^2)+c_7 (-x_3 x_5 + 1/3 (-2 \zeta_6 + 1) x_3 x_6 + (2 \zeta_6 - 1) x_4 x_5 - x_4 x_6)+c_8 ((\zeta_6 + 1) x_5 x_6 + (\zeta_6 - 1) x_6^2) \end{array} \]

The multiplicities shown indicate that we require a submodule \(W \subseteq V_{14}^{\oplus 4}\) such that \( W \cong V_{14}^{\oplus 2} \). Thus, our candidate polynomials have the form \[ \begin{array}{l} c_5 x_1^2+c_6 (x_4^2)+c_7 (x_3 x_5 - x_3 x_6 + 1/3 (-2 \zeta_6 + 1) x_4 x_5 + 1/3 (2 \zeta_6 - 1) x_4 x_6)+c_8 (x_5^2 + (\zeta_6 - 1) x_5 x_6 - \zeta_6 x_6^2) \\ c_5 x_1 x_2+c_6 (-x_3 x_4)+c_7 (1/3 (-2 \zeta_6 + 1) x_3 x_5 + 1/3 (2 \zeta_6 - 1) x_3 x_6 - x_4 x_5 + 1/3 (-2 \zeta_6 + 1) x_4 x_6)+c_8 (1/2 (-2 \zeta_6 + 1) x_5^2 + \zeta_6 x_5 x_6 - 1/2 x_6^2) \\ c_5 x_2^2+c_6 (x_3^2)+c_7 (-x_3 x_5 + 1/3 (-2 \zeta_6 + 1) x_3 x_6 + (2 \zeta_6 - 1) x_4 x_5 - x_4 x_6)+c_8 ((\zeta_6 + 1) x_5 x_6 + (\zeta_6 - 1) x_6^2) \\ c_9 x_1^2+c_{10} (x_4^2)+c_{11} (x_3 x_5 - x_3 x_6 + 1/3 (-2 \zeta_6 + 1) x_4 x_5 + 1/3 (2 \zeta_6 - 1) x_4 x_6)+c_{12} (x_5^2 + (\zeta_6 - 1) x_5 x_6 - \zeta_6 x_6^2) \\ c_9 x_1 x_2+c_{10} (-x_3 x_4)+c_{11} (1/3 (-2 \zeta_6 + 1) x_3 x_5 + 1/3 (2 \zeta_6 - 1) x_3 x_6 - x_4 x_5 + 1/3 (-2 \zeta_6 + 1) x_4 x_6)+c_{12} (1/2 (-2 \zeta_6 + 1) x_5^2 + \zeta_6 x_5 x_6 - 1/2 x_6^2) \\ c_9 x_2^2+c_{10} (x_3^2)+c_{11} (-x_3 x_5 + 1/3 (-2 \zeta_6+ 1) x_3 x_6 + (2 \zeta_6 - 1) x_4 x_5 - x_4 x_6)+c_{12} ((\zeta_6+ 1) x_5 x_6 + (\zeta_6 - 1) x_6^2) \end{array} \]

Assuming that \(c_5\) and \(c_{10}\) are nonzero, after replacing these quadrics with appropriate linear combinations, we may assume that \(c_5=1\), \(c_{6}=0\), \(c_9=0\), and \(c_{10}=1\). Furthermore, assuming that \(c_7\) is nonzero, by scaling \( x_5 \) and \(x_6\), we may assume that \(c_7=1\). This yields \[ \begin{array}{l} x_1^2+x_3 x_5 - x_3 x_6 + 1/3 (-2 \zeta_6 + 1) x_4 x_5 + 1/3 (2 \zeta_6 - 1) x_4 x_6+c_8 (x_5^2 + (\zeta_6 - 1) x_5 x_6 - \zeta_6 x_6^2) \\ x_1 x_2+1/3 (-2 \zeta_6+ 1) x_3 x_5 + 1/3 (2 \zeta_6 - 1) x_3 x_6 - x_4 x_5 + 1/3 (-2 \zeta_6 + 1) x_4 x_6+c_8 (1/2 (-2 \zeta_6 + 1) x_5^2 + t x_5 x_6 - 1/2 x_6^2) \\ x_2^2+-x_3 x_5 + 1/3 (-2 \zeta_6 + 1) x_3 x_6 + (2 \zeta_6 - 1) x_4 x_5 - x_4 x_6+c_8 ((\zeta_6 + 1) x_5 x_6 + (\zeta_6 - 1) x_6^2) \\ x_4^2+c_{11} (x_3 x_5 - x_3 x_6 + 1/3 (-2 \zeta_6 + 1) x_4 x_5 + 1/3 (2 \zeta_6- 1) x_4 x_6)+c_{12} (x_5^2 + (\zeta_6 - 1) x_5 x_6 - \zeta_6 x_6^2) \\ -x_3 x_4+c_{11} (1/3 (-2\zeta_6 + 1) x_3 x_5 + 1/3 (2 \zeta_6 - 1) x_3 x_6 - x_4 x_5 + 1/3 (-2 \zeta_6 + 1) x_4 x_6)+c_{12} (1/2 (-2 \zeta_6 + 1) x_5^2 + \zeta_6 x_5 x_6 - 1/2 x_6^2) \\ x_3^2+c_{11} (-x_3 x_5 + 1/3 (-2 \zeta_6 + 1) x_3 x_6 + (2 \zeta_6 - 1) x_4 x_5 - x_4 x_6)+c_{12} ((\zeta_6 + 1) x_5 x_6 + (\zeta_6 - 1) x_6^2) \end{array} \] For generic values of \(c_8, c_{11}, c_{12}\), the intersection of these 10 quadrics in \(\mathbb{P}^6\) is empty. Here is an example showing this: 
> K<z_6>:=CyclotomicField(6);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_5:=1;
> c_6:=0;
> c_7:=1;
> c_8:=54+12*z_6;
> c_9:=0;
> c_10:=1;
> c_11:=24+19*z_6;
> c_12:=7+71*z_6;
> X:=Scheme(P6,[
> x_0^2+x_3*x_5 + z_6*x_3*x_6 + (-z_6 + 1)*x_4*x_6,
> x_1*x_3 - x_2*x_4 + x_1*x_5 + 1/3*(2*z_6 - 1)*x_1*x_6 + 1/3*(-2*z_6 + 1)*x_2*x_5\
 + 1/3*(2*z_6 - 1)*x_2*x_6,
> x_1*x_4+1/6*(-2*z_6 + 1)*x_1*x_5 + 1/6*(2*z_6 - 1)*x_1*x_6 - 1/2*x_2*x_5 + 1/2*x\
_2*x_6,
> x_2*x_3+ 1/2*(-2*z_6 + 1)*x_1*x_5 + 1/2*x_1*x_6 - 1/2*x_2*x_5 + 1/6*(-2*z_6 + 1)\
*x_2*x_6,
> c_5*x_1^2+c_6*(x_4^2)+c_7*(x_3*x_5 - x_3*x_6 + 1/3*(-2*z_6 + 1)*x_4*x_5 + 1/3*\
(2*z_6 - 1)*x_4*x_6)+c_8*(x_5^2 + (z_6 - 1)*x_5*x_6 - z_6*x_6^2),
> c_5*x_1*x_2+c_6*(-x_3*x_4)+c_7*(1/3*(-2*z_6 + 1)*x_3*x_5 + 1/3*(2*z_6 - 1)*x_3*x\
_6 - x_4*x_5 + 1/3*(-2*z_6 + 1)*x_4*x_6)+c_8*(1/2*(-2*z_6 + 1)*x_5^2 + z_6*x_5*x_6 -\
 1/2*x_6^2),
> c_5*x_2^2+c_6*(x_3^2)+c_7*(-x_3*x_5 + 1/3*(-2*z_6 + 1)*x_3*x_6 + (2*z_6 - 1)*x_4\
*x_5 - x_4*x_6)+c_8*((z_6 + 1)*x_5*x_6 + (z_6 - 1)*x_6^2),
> c_9*x_1^2+c_10*(x_4^2)+c_11*(x_3*x_5 - x_3*x_6 + 1/3*(-2*z_6 + 1)*x_4*x_5 + 1/\
3*(2*z_6 - 1)*x_4*x_6)+c_12*(x_5^2 + (z_6 - 1)*x_5*x_6 - z_6*x_6^2),
> c_9*x_1*x_2+c_10*(-x_3*x_4)+c_11*(1/3*(-2*z_6 + 1)*x_3*x_5 + 1/3*(2*z_6 - 1)*x_3\
*x_6 - x_4*x_5 + 1/3*(-2*z_6 + 1)*x_4*x_6)+c_12*(1/2*(-2*z_6 + 1)*x_5^2 + z_6*x_5*x_\
6 - 1/2*x_6^2),
> c_9*x_2^2+c_10*(x_3^2)+c_11*(-x_3*x_5 + 1/3*(-2*z_6 + 1)*x_3*x_6 + (2*z_6 - 1)*x\
_4*x_5 - x_4*x_6)+c_12*((z_6 + 1)*x_5*x_6 + (z_6 - 1)*x_6^2)
> ]);
> 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(6);

i3 : z_6=K_0;

i4 : S=K[c_8,c_11,c_12,Degrees=>{0,0,0}];

i5 : c_5=1;

i6 : c_6=0;

i7 : c_7=1;

i8 : c_9=0;

i9 : c_10=1;

i10 : T=S[x_0..x_6];

i11 : I=ideal({x_0^2+x_3*x_5 + z_6*x_3*x_6 + (-z_6 + 1)*x_4*x_6,
      x_1*x_3 - x_2*x_4 + x_1*x_5 + 1/3*(2*z_6 - 1)*x_1*x_6 + 1/3*(-2*z_6 + 1)*x_2*x_5 + 1/3*(2*z_6 - 1)*x_2*x_6,
      x_1*x_4+1/6*(-2*z_6 + 1)*x_1*x_5 + 1/6*(2*z_6 - 1)*x_1*x_6 - 1/2*x_2*x_5 + 1/2*x_2*x_6,
      x_2*x_3+ 1/2*(-2*z_6 + 1)*x_1*x_5 + 1/2*x_1*x_6 - 1/2*x_2*x_5 + 1/6*(-2*z_6 + 1)*x_2*x_6,
      c_5*x_1^2+c_6*(x_4^2)+c_7*(x_3*x_5 - x_3*x_6 + 1/3*(-2*z_6 + 1)*x_4*x_5 + 1/3*(2*z_6 - 1)*x_4*x_6)+c_8*(x_5^2 + (z_6 - 1)*x_5*x_6 - z_6*x_6^2),
      c_5*x_1*x_2+c_6*(-x_3*x_4)+c_7*(1/3*(-2*z_6 + 1)*x_3*x_5 + 1/3*(2*z_6 - 1)*x_3*x_6 - x_4*x_5 + 1/3*(-2*z_6 + 1)*x_4*x_6)+c_8*(1/2*(-2*z_6 + 1)*x_5^2 + z_6*x_5*x_6 - 1/2*x_6^2),
      c_5*x_2^2+c_6*(x_3^2)+c_7*(-x_3*x_5 + 1/3*(-2*z_6 + 1)*x_3*x_6 + (2*z_6 - 1)*x_4*x_5 - x_4*x_6)+c_8*((z_6 + 1)*x_5*x_6 + (z_6 - 1)*x_6^2),
      c_9*x_1^2+c_10*(x_4^2)+c_11*(x_3*x_5 - x_3*x_6 + 1/3*(-2*z_6 + 1)*x_4*x_5 + 1/3*(2*z_6 - 1)*x_4*x_6)+c_12*(x_5^2 + (z_6 - 1)*x_5*x_6 - z_6*x_6^2),
      c_9*x_1*x_2+c_10*(-x_3*x_4)+c_11*(1/3*(-2*z_6 + 1)*x_3*x_5 + 1/3*(2*z_6 - 1)*x_3*x_6 - x_4*x_5 + 1/3*(-2*z_6 + 1)*x_4*x_6)+c_12*(1/2*(-2*z_6 + 1)*x_5^2 + z_6*x_5*x_6 - 1/2*x_6^2),
      c_9*x_2^2+c_10*(x_3^2)+c_11*(-x_3*x_5 + 1/3*(-2*z_6 + 1)*x_3*x_6 + (2*z_6 - 1)*x_4*x_5 - x_4*x_6)+c_12*((z_6 + 1)*x_5*x_6 + (z_6 - 1)*x_6^2)});

o11 : Ideal of T

i12 : L=flatten entries gens gb(I,DegreeLimit=>4);

i13 : tally apply(L, i -> degree i)

o13 = Tally{{2, 0} => 10}
            {3, 0} => 14

o13 : Tally

i14 : for i from 0 to #L-1 do (if degree(L_i) == {3,0} then print toString(L_i) << endl)
(c_8-(2/3)*c_11+1/3)*x_3*x_5*x_6+(((1/3)*ww_6-2/3)*c_8+(-(2/9)*ww_6+4/9)*c_11+(1/9)*ww_6-2/9)*x_3*x_6^2+((ww_6-1)*c_8+(-(2/3)*ww_6+2/3)*c_11+(1/3)*ww_6-1/3)*x_4*x_6^2+(-(1/3)*ww_6*c_8+(2/3)*ww_6*c_12)*x_6^3
(c_11-3*c_12-1/2)*x_2*x_5*x_6+((ww_6-1)*c_11+(-3*ww_6+3)*c_12-(1/2)*ww_6+1/2)*x_1*x_6^2+(ww_6*c_11-3*ww_6*c_12-(1/2)*ww_6)*x_2*x_6^2
(c_8-2*c_12)*x_5^3+((2*ww_6-1)*c_8+(-4*ww_6+2)*c_12)*x_5^2*x_6+(-(4/3)*ww_6*c_8+(8/9)*ww_6*c_11-(4/9)*ww_6)*x_3*x_6^2+(((4/3)*ww_6-8/3)*c_8+(-(8/9)*ww_6+16/9)*c_11+(4/9)*ww_6-8/9)*x_4*x_6^2+(-c_8+2*c_12)*x_5*x_6^2+((-(10/9)*ww_6+5/9)*c_8+((20/9)*ww_6-10/9)*c_12)*x_6^3
(c_11+1/2)*x_4*x_5^2+(((2/3)*ww_6-2/3)*c_11+(1/3)*ww_6-1/3)*x_3*x_5*x_6+(((1/2)*ww_6+1/2)*c_8+((5/3)*ww_6-1/3)*c_11+(7/6)*ww_6+1/6)*x_4*x_5*x_6+(((1/4)*ww_6-1/4)*c_8+(-(1/2)*ww_6+1/2)*c_12)*x_5^2*x_6+((-(1/6)*ww_6+1/3)*c_8+(-(5/9)*ww_6+4/9)*c_11-(7/18)*ww_6+4/9)*x_3*x_6^2+((ww_6-1/2)*c_8-(2/3)*c_11+(2/3)*ww_6-2/3)*x_4*x_6^2+((-(1/3)*ww_6+1/6)*c_8+((2/3)*ww_6-1/3)*c_12)*x_5*x_6^2+(((1/12)*ww_6-1/12)*c_8+(-(1/6)*ww_6+1/6)*c_12)*x_6^3
(c_8+2/3)*x_4*x_5^2+(((4/9)*ww_6-4/9)*c_11+(2/9)*ww_6-2/9)*x_3*x_5*x_6+(((4/3)*ww_6-2/3)*c_8+((4/9)*ww_6+4/9)*c_11+(10/9)*ww_6-2/9)*x_4*x_5*x_6+((-(1/3)*ww_6+1/3)*c_8+((2/3)*ww_6-2/3)*c_12)*x_5^2*x_6+((-(4/9)*ww_6+4/9)*c_11-(2/9)*ww_6+2/9)*x_3*x_6^2+(-(1/3)*c_8+((4/9)*ww_6-4/9)*c_11+(2/9)*ww_6-4/9)*x_4*x_6^2+(((4/9)*ww_6-2/9)*c_8+(-(8/9)*ww_6+4/9)*c_12)*x_5*x_6^2+((-(1/9)*ww_6-1/9)*c_8+((2/9)*ww_6+2/9)*c_12)*x_6^3
(c_11+1/2)*x_3*x_5^2+(((2/3)*ww_6-4/3)*c_11+(1/3)*ww_6-2/3)*x_3*x_5*x_6+(((3/2)*ww_6-3/2)*c_8+(ww_6-1)*c_11+(3/2)*ww_6-3/2)*x_4*x_5*x_6+(-(1/2)*ww_6*c_8+(-(1/3)*ww_6+1/3)*c_11-(1/2)*ww_6+1/6)*x_3*x_6^2+(-(3/2)*c_8+(-(2/3)*ww_6+1/3)*c_11-(1/3)*ww_6-5/6)*x_4*x_6^2+((1/2)*ww_6*c_8-ww_6*c_12)*x_5*x_6^2+((-(1/6)*ww_6-1/6)*c_8+((1/3)*ww_6+1/3)*c_12)*x_6^3
(c_8+2/3)*x_3*x_5^2+(((4/9)*ww_6-8/9)*c_11+(2/9)*ww_6-4/9)*x_3*x_5*x_6+(((4/3)*ww_6-4/3)*c_11+(2/3)*ww_6-2/3)*x_4*x_5*x_6+(((2/3)*ww_6-1/3)*c_8+(-(8/9)*ww_6+4/9)*c_11)*x_3*x_6^2+(((2/3)*ww_6+2/3)*c_8+(-(8/9)*ww_6-8/9)*c_11)*x_4*x_6^2+(-(2/3)*ww_6*c_8+(4/3)*ww_6*c_12)*x_5*x_6^2
(c_12+1/3)*x_2*x_5^2+((-(2/3)*ww_6+2/3)*c_11-(1/3)*ww_6+1/3)*x_1*x_5*x_6+(((2/3)*ww_6-4/3)*c_12+(2/9)*ww_6-4/9)*x_2*x_5*x_6+(((4/9)*ww_6-2/9)*c_11+(-(2/3)*ww_6+4/3)*c_12+1/3)*x_1*x_6^2+((2/9)*ww_6*c_11+(-(4/3)*ww_6+1/3)*c_12-(1/3)*ww_6+1/9)*x_2*x_6^2
(c_11+1/2)*x_2*x_5^2+((-ww_6+1)*c_11+(-3*ww_6+3)*c_12-(3/2)*ww_6+3/2)*x_1*x_5*x_6+((2*ww_6-4)*c_12+(2/3)*ww_6-4/3)*x_2*x_5*x_6+(((4/3)*ww_6+1/3)*c_11+(-2*ww_6+1)*c_12+1/2)*x_1*x_6^2+((-(1/3)*ww_6+1)*c_11+(-ww_6-2)*c_12-(1/2)*ww_6-1/6)*x_2*x_6^2
(c_8+2/3)*x_2*x_5^2+((-2*ww_6+2)*c_8-(4/3)*ww_6+4/3)*x_1*x_5*x_6+(((2/3)*ww_6-4/3)*c_8+(4/9)*ww_6-8/9)*x_2*x_5*x_6+(((2/3)*ww_6+2/3)*c_8+(4/9)*ww_6+4/9)*x_1*x_6^2+((-(2/3)*ww_6+1/3)*c_8-(4/9)*ww_6+2/9)*x_2*x_6^2
(c_12+1/3)*x_1*x_5^2+((-(4/9)*ww_6-4/9)*c_11-(2/9)*ww_6-2/9)*x_2*x_4*x_6+(((4/9)*ww_6+4/9)*c_11+((4/3)*ww_6-2/3)*c_12+(2/3)*ww_6)*x_1*x_5*x_6+((-(4/3)*ww_6+4/3)*c_12-(4/9)*ww_6+4/9)*x_2*x_5*x_6+(-(4/9)*c_11+((4/3)*ww_6-1/3)*c_12+(4/9)*ww_6-1/3)*x_1*x_6^2+(((4/9)*ww_6-8/9)*c_11+(4/3)*c_12+(2/9)*ww_6)*x_2*x_6^2
(c_11+1/2)*x_1*x_5^2+(((2/3)*ww_6+2/3)*c_11+(1/3)*ww_6+1/3)*x_2*x_4*x_6+(((4/3)*ww_6-2/3)*c_11+(ww_6+1)*c_12+ww_6)*x_1*x_5*x_6+((-ww_6+1)*c_12-(1/3)*ww_6+1/3)*x_2*x_5*x_6+(-(1/3)*c_11+(ww_6-1)*c_12+(1/3)*ww_6-1/2)*x_1*x_6^2+((ww_6-1)*c_12+(1/3)*ww_6-1/3)*x_2*x_6^2
(c_8+2/3)*x_1*x_5^2+(2*ww_6*c_8+(4/3)*ww_6)*x_1*x_5*x_6+((-(2/3)*ww_6+2/3)*c_8-(4/9)*ww_6+4/9)*x_2*x_5*x_6+(((2/3)*ww_6-1)*c_8+(4/9)*ww_6-2/3)*x_1*x_6^2+(((2/3)*ww_6-2/3)*c_8+(4/9)*ww_6-4/9)*x_2*x_6^2
(c_11+1/2)*x_2*x_4*x_5+(ww_6*c_11+(1/2)*ww_6)*x_2*x_4*x_6+((-(1/6)*ww_6-1/6)*c_11-(1/12)*ww_6-1/12)*x_1*x_5*x_6+((-(1/2)*ww_6+1/2)*c_12-(1/6)*ww_6+1/6)*x_2*x_5*x_6+((1/6)*c_11-(1/2)*ww_6*c_12-(1/6)*ww_6+1/12)*x_1*x_6^2+((-(1/18)*ww_6+1/9)*c_11+((2/3)*ww_6-5/6)*c_12+(7/36)*ww_6-2/9)*x_2*x_6^2

The leading coefficients of the generators above suggest the choices \( c_8 = -2/3\), \(c_{11}=-1/2\), and \(c_{12}=-1/3\). In the next section we show that these choices yield equations of the desired curve.

Checking the equations in Magma

We check that our equations we obtained give a smooth genus 7 curve with the desired automorphisms. 
> K<z_6>:=CyclotomicField(6);
> GL7K:=GeneralLinearGroup(7,K);
> A:=Matrix([[z_6-1,0,0,0,0,0,0],
> [0,0,-z_6+1,0,0,0,0],
> [0,-z_6,-1,0,0,0,0],
> [0,0,0,-1,z_6,0,0],
> [0,0,0,z_6-1,0,0,0],
> [0,0,0,0,0,z_6-1,-z_6],
> [0,0,0,0,0,0,1]]);
> B:=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,-1,0,0],
> [0,0,0,1,0,0,0],
> [0,0,0,0,0,z_6-1,-z_6],
> [0,0,0,0,0,-z_6,-z_6+1]]);
> Order(A);
3
> Order(B);
4
> IdentifyGroup(sub<GL7K | A,B>);
<48, 32>
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_5:=1;
> c_6:=0;
> c_9:=0;
> c_7:=1;
> c_8:=-2/3;
> c_10:=1;
> c_11:=-1/2;
> c_12:=-1/3;
> X:=Scheme(P6,[
> x_0^2+x_3*x_5 + z_6*x_3*x_6 + (-z_6 + 1)*x_4*x_6,
> x_1*x_3 - x_2*x_4 + x_1*x_5 + 1/3*(2*z_6 - 1)*x_1*x_6 + 1/3*(-2*z_6 + 1)*x_2*x_5\
 + 1/3*(2*z_6 - 1)*x_2*x_6,
> x_1*x_4+1/6*(-2*z_6 + 1)*x_1*x_5 + 1/6*(2*z_6 - 1)*x_1*x_6 - 1/2*x_2*x_5 + 1/2*x\
_2*x_6,
> x_2*x_3+ 1/2*(-2*z_6 + 1)*x_1*x_5 + 1/2*x_1*x_6 - 1/2*x_2*x_5 + 1/6*(-2*z_6 + 1)\
*x_2*x_6,
> c_5*x_1^2+c_6*(x_4^2)+c_7*(x_3*x_5 - x_3*x_6 + 1/3*(-2*z_6 + 1)*x_4*x_5 + 1/3*\
(2*z_6 - 1)*x_4*x_6)+c_8*(x_5^2 + (z_6 - 1)*x_5*x_6 - z_6*x_6^2),
> c_5*x_1*x_2+c_6*(-x_3*x_4)+c_7*(1/3*(-2*z_6 + 1)*x_3*x_5 + 1/3*(2*z_6 - 1)*x_3*x\
_6 - x_4*x_5 + 1/3*(-2*z_6 + 1)*x_4*x_6)+c_8*(1/2*(-2*z_6 + 1)*x_5^2 + z_6*x_5*x_6 -\
 1/2*x_6^2),
> c_5*x_2^2+c_6*(x_3^2)+c_7*(-x_3*x_5 + 1/3*(-2*z_6 + 1)*x_3*x_6 + (2*z_6 - 1)*x_4\
*x_5 - x_4*x_6)+c_8*((z_6 + 1)*x_5*x_6 + (z_6 - 1)*x_6^2),
> c_9*x_1^2+c_10*(x_4^2)+c_11*(x_3*x_5 - x_3*x_6 + 1/3*(-2*z_6 + 1)*x_4*x_5 + 1/\
3*(2*z_6 - 1)*x_4*x_6)+c_12*(x_5^2 + (z_6 - 1)*x_5*x_6 - z_6*x_6^2),
> c_9*x_1*x_2+c_10*(-x_3*x_4)+c_11*(1/3*(-2*z_6 + 1)*x_3*x_5 + 1/3*(2*z_6 - 1)*x_3\
*x_6 - x_4*x_5 + 1/3*(-2*z_6 + 1)*x_4*x_6)+c_12*(1/2*(-2*z_6 + 1)*x_5^2 + z_6*x_5*x_\
6 - 1/2*x_6^2),
> c_9*x_2^2+c_10*(x_3^2)+c_11*(-x_3*x_5 + 1/3*(-2*z_6 + 1)*x_3*x_6 + (2*z_6 - 1)*x\
_4*x_5 - x_4*x_6)+c_12*((z_6 + 1)*x_5*x_6 + (z_6 - 1)*x_6^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
(z_6 - 1)*x_0
-z_6*x_2
(-z_6 + 1)*x_1 - x_2
-x_3 + (z_6 - 1)*x_4
z_6*x_3
(z_6 - 1)*x_5
-z_6*x_5 + x_6
and inverse
-z_6*x_0
-x_1 + z_6*x_2
(z_6 - 1)*x_1
(-z_6 + 1)*x_4
-z_6*x_3 - x_4
-z_6*x_5
(-z_6 + 1)*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(6);

i3 : z_6=K_0;

i4 : c_5=1;

i5 : c_6=0;

i6 : c_7=1;

i7 : c_8=-2/3;

i8 : c_9=0;

i9 : c_10=1;

i10 : c_11=-1/2;

i11 : c_12=-1/3;

i12 : T=K[x_0..x_6];

i13 : I=ideal({x_0^2+x_3*x_5 + z_6*x_3*x_6 + (-z_6 + 1)*x_4*x_6,
      x_1*x_3 - x_2*x_4 + x_1*x_5 + 1/3*(2*z_6 - 1)*x_1*x_6 + 1/3*(-2*z_6 + 1)*x_2*x_5 + 1/3*(2*z_6 - 1)*x_2*x_6,
      x_1*x_4+1/6*(-2*z_6 + 1)*x_1*x_5 + 1/6*(2*z_6 - 1)*x_1*x_6 - 1/2*x_2*x_5 + 1/2*x_2*x_6,
      x_2*x_3+ 1/2*(-2*z_6 + 1)*x_1*x_5 + 1/2*x_1*x_6 - 1/2*x_2*x_5 + 1/6*(-2*z_6 + 1)*x_2*x_6,
      c_5*x_1^2+c_6*(x_4^2)+c_7*(x_3*x_5 - x_3*x_6 + 1/3*(-2*z_6 + 1)*x_4*x_5 + 1/3*(2*z_6 - 1)*x_4*x_6)+c_8*(x_5^2 + (z_6 - 1)*x_5*x_6 - z_6*x_6^2),
      c_5*x_1*x_2+c_6*(-x_3*x_4)+c_7*(1/3*(-2*z_6 + 1)*x_3*x_5 + 1/3*(2*z_6 - 1)*x_3*x_6 - x_4*x_5 + 1/3*(-2*z_6 + 1)*x_4*x_6)+c_8*(1/2*(-2*z_6 + 1)*x_5^2 + z_6*x_5*x_6 - 1/2*x_6^2),
      c_5*x_2^2+c_6*(x_3^2)+c_7*(-x_3*x_5 + 1/3*(-2*z_6 + 1)*x_3*x_6 + (2*z_6 - 1)*x_4*x_5 - x_4*x_6)+c_8*((z_6 + 1)*x_5*x_6 + (z_6 - 1)*x_6^2),
      c_9*x_1^2+c_10*(x_4^2)+c_11*(x_3*x_5 - x_3*x_6 + 1/3*(-2*z_6 + 1)*x_4*x_5 + 1/3*(2*z_6 - 1)*x_4*x_6)+c_12*(x_5^2 + (z_6 - 1)*x_5*x_6 - z_6*x_6^2),
      c_9*x_1*x_2+c_10*(-x_3*x_4)+c_11*(1/3*(-2*z_6 + 1)*x_3*x_5 + 1/3*(2*z_6 - 1)*x_3*x_6 - x_4*x_5 + 1/3*(-2*z_6 + 1)*x_4*x_6)+c_12*(1/2*(-2*z_6 + 1)*x_5^2 + z_6*x_5*x_6 - 1/2*x_6^2),
      c_9*x_2^2+c_10*(x_3^2)+c_11*(-x_3*x_5 + 1/3*(-2*z_6 + 1)*x_3*x_6 + (2*z_6 - 1)*x_4*x_5 - x_4*x_6)+c_12*((z_6 + 1)*x_5*x_6 + (z_6 - 1)*x_6^2)});

o13 : Ideal of T

i14 : betti res I

             0  1  2  3  4 5
o14 = total: 1 10 25 25 10 1
          0: 1  .  .  .  . .
          1: . 10 16  9  . .
          2: .  .  9 16 10 .
          3: .  .  .  .  . 1

o14 : BettiTally
By [Schreyer1986], this Betti table implies that the curve has a \( g^{2}_{6} \).