Fordham University

Finding equations of a 1-parameter family of genus 7 Riemann surfaces with 36 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 1-parameter family of genus 4 Riemann surfaces with automorphism group (36,10) in the GAP library of small groups. The quotient of any surface in this family has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,2,6).

We use Magma to compute equations of one member of this family, and give a conjectural description of this family.

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 V2.21-4     Sat Nov 14 2015 09:47:57 on ace-math01 [Seed = 874851192]
Type ? for help.  Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> G:=SmallGroup(36,10);

> MatrixG,MatrixSKG,Q,C:=RunExample(G,7,[2,2,2,6]);
Set seed to 0.


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


-----------------------------------
Class |   1  2  3  4  5  6  7  8  9
Size  |   1  3  3  9  2  2  4  6  6
Order |   1  2  2  2  3  3  3  6  6
-----------------------------------
p  =  2   1  1  1  1  5  6  7  5  6
p  =  3   1  2  3  4  1  1  1  3  2
-----------------------------------
X.1   +   1  1  1  1  1  1  1  1  1
X.2   +   1 -1  1 -1  1  1  1  1 -1
X.3   +   1  1 -1 -1  1  1  1 -1  1
X.4   +   1 -1 -1  1  1  1  1 -1 -1
X.5   +   2  2  0  0  2 -1 -1  0 -1
X.6   +   2 -2  0  0  2 -1 -1  0  1
X.7   +   2  0  2  0 -1  2 -1 -1  0
X.8   +   2  0 -2  0 -1  2 -1  1  0
X.9   +   4  0  0  0 -2 -2  1  0  0



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

[2]     Order 2       Length 3      
        Rep G.2

[3]     Order 2       Length 3      
        Rep G.1

[4]     Order 2       Length 9      
        Rep G.1 * G.2

[5]     Order 3       Length 2      
        Rep G.3

[6]     Order 3       Length 2      
        Rep G.4

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

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

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


Surface kernel generators:  [ G.1 * G.2 * G.3^2, G.1 * G.2 * G.4^2, G.2, G.2 * 
G.3^2 * G.4 ]
Is hyperelliptic?  false
Is cyclic trigonal?  false
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1      =[ 0, 1, 0, 0, 0, 1, 0, 0, 1 ]
H^0(C,K) =[ 0, 1, 0, 0, 0, 1, 0, 0, 1 ]
I_2      =[ 2, 0, 0, 0, 1, 0, 1, 0, 1 ]
S_2      =[ 3, 0, 0, 1, 3, 0, 2, 1, 3 ]
H^0(C,2K)=[ 1, 0, 0, 1, 2, 0, 1, 1, 2 ]
I_3      =[ 1, 4, 1, 2, 1, 5, 3, 2, 6 ]
S_3      =[ 1, 6, 2, 3, 2, 7, 5, 4, 9 ]
H^0(C,3K)=[ 0, 2, 1, 1, 1, 2, 2, 2, 3 ]
I2timesS1=[ 1, 4, 1, 2, 2, 6, 4, 3, 8 ]
Is clearly not generated by quadrics? false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 36 and degree 12
[
    [ 1  0  0  0  0  0  0]
    [ 0  0 -1  0  0  0  0]
    [ 0 -1  0  0  0  0  0]
    [ 0  0  0 -1 -1  0  0]
    [ 0  0  0  0  1  0  0]
    [ 0  0  0  1  1  0 -1]
    [ 0  0  0 -1  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  0  0  1  0]
    [ 0  0  0 -1 -1 -1  1]
    [ 0  0  0  1  0  0  0]
    [ 0  0  0  0  0  0  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  0  0  1  0]
    [ 0  0  0 -1 -1 -1  1]
    [ 0  0  0 -1  0 -1  0]
    [ 0  0  0 -1 -1  0  0],

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

    [-1  0  0  0  0  0  0]
    [ 0  1 -1  0  0  0  0]
    [ 0  1  0  0  0  0  0]
    [ 0  0  0 -1 -1  0  0]
    [ 0  0  0  1  0  0  0]
    [ 0  0  0  0  0  0  1]
    [ 0  0  0 -1 -1 -1  1]
]
Finding quadrics:
I2 contains a 2-dimensional subspace of CharacterRow 1
Dimension 3
Multiplicity 3
[
    x_0^2,
    x_1^2 + x_1*x_2 + x_2^2,
    x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 
        + x_5^2 + 1/2*x_5*x_6 + x_6^2
]
I2 contains a 2-dimensional subspace of CharacterRow 5
Dimension 6
Multiplicity 3
[
    x_0*x_1,
    x_0*x_2,
    x_1^2 - x_2^2,
    x_1*x_2 + 1/2*x_2^2,
    x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x_4*x_6 + x_5^2 + x_5*x_6,
    x_3*x_4 + 1/2*x_3*x_6 - 1/2*x_4^2 - 1/2*x_4*x_5 + 1/2*x_5*x_6 + 1/2*x_6^2
]
I2 contains a 2-dimensional subspace of CharacterRow 7
Dimension 4
Multiplicity 2
[
    x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6,
    x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6,
    x_3^2 - x_3*x_4 + x_3*x_6 + x_4^2 + x_4*x_6 - x_5^2 - x_5*x_6,
    x_3*x_5 + 1/2*x_3*x_6 - 1/2*x_4*x_5 + 1/2*x_4*x_6 - 1/2*x_5^2 + 1/2*x_6^2
]
I2 contains a 4-dimensional subspace of CharacterRow 9
Dimension 12
Multiplicity 3
[
    x_0*x_3,
    x_0*x_4,
    x_0*x_5,
    x_0*x_6,
    x_1*x_3 + x_2*x_4 + x_2*x_5 + x_2*x_6,
    x_1*x_4 - x_2*x_3 + x_2*x_4 + x_2*x_5,
    x_1*x_5 + x_2*x_5 + x_2*x_6,
    x_1*x_6 - x_2*x_5,
    x_3^2 - x_4^2 - 2*x_4*x_5 - 2*x_4*x_6 - x_5^2 - 2*x_5*x_6 - x_6^2,
    x_3*x_4 - 1/2*x_4^2 - x_4*x_5 - x_5*x_6 - 1/2*x_6^2,
    x_3*x_5 - x_4*x_5 - x_4*x_6 - 1/2*x_5^2 - 1/2*x_6^2,
    x_3*x_6 + x_4*x_5 + 1/2*x_6^2
]

The first isotypical subspace corresponds to the character \( \chi_1\) in the character table shown above. Let \(f_1, f_2, f_3\) be the three polynomials shown. Then we seek polynomials of the form 


    c_1 f_1 + c_2 f_2 + c_3 f_3
    c_4 f_1 + c_5 f_2 + c_6 f_3

Assume that \((c_1,c_2)\) and \((c_4,c_5)\) are linearly independent. After row reducing, we may assume that \((c_1,c_2)=(1,0)\) and \((c_4,c_5)=(0,1).\) Thus we obtain the polynomials 

x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2)

The second isotypical subspace corresponds to the character \( \chi_{5}\) in the character table shown above. Note that the matrix surface kernel generators have a block diagonal form with blocks of size \(1 \times 1\), \(2 \times 2\), and \(4 \times 4\). We therefore let the first two polynomials shown here generate one copy of \(V_{10}\) and use the FindParallelBases function to find ordered bases of the span of the third and fourth polynomials and the span of the fifth and sixth polynomials such that the action of \(G\) is given by the same matrices relative to all three ordered bases. 


> K<z_36>:=CyclotomicField(36);
> GL7K:=Parent(MatrixG[1]);
> MatrixG:=sub<GL7K | MatrixG>;
> FindParallelBases(MatrixG, [Q[2][1],Q[2][2]],[Q[2][3],Q[2][4]]);
[ x_1^2 + 2*x_1*x_2]
[-2*x_1*x_2 - x_2^2]
> FindParallelBases(MatrixG, [Q[2][1],Q[2][2]],[Q[2][5],Q[2][6]]);
[     -2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*x_5 - x_5*x_6 - x_6^2]
[x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x_4*x_6 + x_5^2 + x_5*x_6]
This yields polynomials of the form 

c_7*(x_0*x_1) + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*x_5 - x_5*x_6 - x_6^2),
c_7*(x_0*x_2) +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x_4*x_6 + x_5^2 + x_5*x_6),
Assume that \(c_7\) is nonzero. Then after dividing by \(c_7\), we may assume that \(c_7=1\).

The third isotypical subspace corresponds to the character \( \chi_7\) in the character table shown above. Again we use the FindParallelBases function to find two ordered bases such that the action of \(G\) is given by the same matrices relative to both ordered bases. 


> FindParallelBases(MatrixG, [Q[3][1],Q[3][2]],[Q[3][3],Q[3][4]]);             
[x_3^2 - x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2
    - x_5*x_6 + x_6^2]
[2*x_3^2 - 2*x_3*x_4 - 2*x_3*x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2
    - 2*x_5*x_6 - x_6^2]
This yields polynomials of the form 

c_10*(x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6) + c_11*(x_3^2 - x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_5*x_6 + x_6^2),
c_10*(x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6) + c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
Assume that \(c_{10}\) is nonzero. Then after dividing by \(c_{10}\), we may assume that \(c_{10}=1\).

The fourth isotypical subspace corresponds to the character \( \chi_{9}\) in the character table shown above. Again we use the FindParallelBases function to find three ordered bases such that the action of \(G\) is given by the same matrices relative to all three ordered bases. 


> FindParallelBases(MatrixG, [Q[4][1],Q[4][2],Q[4][3],Q[4][4]],[Q[4][5],Q[4][6],Q[4][7],Q[4][8]]);
[x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6]
[                    x_1*x_3 - x_1*x_5 + x_2*x_4]
[                   -x_1*x_5 - x_2*x_5 - x_2*x_6]
[                    x_1*x_5 + x_1*x_6 + x_2*x_6]
> FindParallelBases(MatrixG, [Q[4][1],Q[4][2],Q[4][3],Q[4][4]],[Q[4][9],Q[4][10],Q[4][11],Q[4][12]]);
[x_3^2 + 2*x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 
    2*x_5^2 - 4*x_5*x_6 - 2*x_6^2]
[2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_4*x_5 - 2*x_5*x_6 - 2*x_6^2]
[2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6]
[-4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2*x_4*x_5 + 4*x_5*x_6 + x_6^2]
This yields polynomials of the form 

c_12*(x_0*x_3) + c_13*(x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6) + c_14*(x_3^2 + 2*x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*x_6 - 2*x_6^2),
c_12*(x_0*x_4) + c_13*(x_1*x_3 - x_1*x_5 + x_2*x_4) + c_14*(2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_4*x_5 - 2*x_5*x_6 - 2*x_6^2),
c_12*(x_0*x_5) + c_13*(-x_1*x_5 - x_2*x_5 - x_2*x_6) + c_14*(2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6),
c_12*(x_0*x_6) + c_13*(x_1*x_5 + x_1*x_6 + x_2*x_6) + c_14*(-4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2*x_4*x_5 + 4*x_5*x_6 + x_6^2)

Assume that \(c_{12},c_{13}, c_{14}\) are nonzero. After dividing by \(c_{14}\), we may assume that \(c_{14}= 1.\) After scaling \(x_0\) we may assume that \(c_{12}=0\), and after scaling \(x_1,x_2\) we may assume that \(c_{13}=1\). We collect all the polynomials listed above and the assumptions we made about the coefficients \(c_i\) to obtain the following candidate polynomials: \[ \begin{array}{l} x_0^2+c_3 (x_3^2 - x_3 x_4 - x_3 x_5 + \frac{1}{2} x_3 x_6 + x_4^2 + \frac{1}{2} x_4 x_5 + \frac{1}{2} x_4 x_6 + x_5^2 + \frac{1}{2} x_5 x_6 + x_6^2),\\ x_1^2 + x_1 x_2 + x_2^2+c_6 (x_3^2 - x_3 x_4 - x_3 x_5 + \frac{1}{2} x_3 x_6 + x_4^2 + \frac{1}{2} x_4 x_5 + \frac{1}{2} x_4 x_6 + x_5^2 + \frac{1}{2} x_5 x_6 + x_6^2)\\ x_0 x_1 + c_8 (x_1^2 + 2 x_1 x_2) + c_9 (-2 x_3 x_4 - x_3 x_6 + x_4^2 + x_4 x_5 - x_5 x_6 - x_6^2),\\ x_0 x_2 +c_8 (-2 x_1 x_2 - x_2^2)+c_9 (x_3^2 - x_3 x_5 + x_3 x_6 - x_4^2 - x_4 x_6 + x_5^2 + x_5 x_6),\\ x_1 x_3 + x_1 x_4 + 2 x_1 x_6 + 2 x_2 x_3 - x_2 x_4 + x_2 x_6 + c_{11} (x_3^2 - x_3 x_4 + 2 x_3 x_5 + 2 x_3 x_6 + x_4^2 - x_4 x_5 + 2 x_4 x_6 - 2 x_5^2 - x_5 x_6 + x_6^2),\\ x_1 x_5 + 2 x_1 x_6 + 2 x_2 x_5 + x_2 x_6+ c_{11} (2 x_3^2 -2 x_3 x_4 - 2 x_3 x_5 + x_3 x_6 + 2 x_4^2 + x_4 x_5 + x_4 x_6 - x_5^2- 2 x_5 x_6 - x_6^2),\\ x_0 x_3 + x_1 x_4 - x_1 x_5 - x_2 x_3 + x_2 x_4 - x_2 x_6 + x_3^2 + 2 x_3 x_4 + 2 x_3 x_5 + 2 x_3 x_6 - 2 x_4^2 - 4 x_4 x_5 - 4 x_4 x_6 - 2 x_5^2 - 4 x_5 x_6 - 2 x_6^2,\\ x_0 x_4 + x_1 x_3 - x_1 x_5 + x_2 x_4 + 2 x_3 x_4 - 2 x_3 x_6 - x_4^2 - 4 x_4 x_5 - 2 x_5 x_6 - 2 x_6^2,\\ x_0 x_5 -x_1 x_5 - x_2 x_5 - x_2 x_6 + 2 x_3^2 - 2 x_3 x_5 + 2 x_3 x_6 - 2 x_4^2 - 2 x_4 x_6 - x_5^2 - 4 x_5 x_6,\\ x_0 x_6 + x_1 x_5 + x_1 x_6 + x_2 x_6 + -4 x_3 x_4 - 2 x_3 x_6 + 2 x_4^2 + 2 x_4 x_5 + 4 x_5 x_6 + x_6^2 \end{array} \] For generic values of \(c_3,c_6, c_{8},c_{9},c_{11}\), the intersection of these 10 quadrics in \(\mathbb{P}^6\) is empty. Here is an example showing this: 
> K:=RationalField();
> c_3:=1;
> c_6:=1;
> c_8:=1;
> c_9:=1;
> c_11:=1;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1\
/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
> x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2\
 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
> x_0*x_1 + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*\
x_5 - x_5*x_6 - x_6^2),
> x_0*x_2 +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x\
_4*x_6 + x_5^2 + x_5*x_6),
> x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6 + c_11*(x_3^2 \
- x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_\
5*x_6 + x_6^2),
> x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6+ c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*\
x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
> x_0*x_3 + x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6 + x_3^2 + 2*x_3*x_\
4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*\
x_6 - 2*x_6^2,
> x_0*x_4 + x_1*x_3 - x_1*x_5 + x_2*x_4 + 2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_\
4*x_5 - 2*x_5*x_6 - 2*x_6^2,
> x_0*x_5 -x_1*x_5 - x_2*x_5 - x_2*x_6 + 2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x\
_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6,
> x_0*x_6 + x_1*x_5 + x_1*x_6 + x_2*x_6 + -4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2\
*x_4*x_5 + 4*x_5*x_6 + 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 : S=QQ[c_3,c_6,c_8,c_9,c_11,Degrees=>{0,0,0,0,0}]

o1 = S

o1 : PolynomialRing

i2 : T=S[x_0..x_6];

i3 : I=ideal({
     x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
     x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
     x_0*x_1 + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*x_5 - x_5*x_6 - x_6^2),
     x_0*x_2 +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x_4*x_6 + x_5^2 + x_5*x_6),
     x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6 + c_11*(x_3^2 - x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_5*x_6 + x_6^2),
     x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6+ c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
     x_0*x_3 + x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6 + x_3^2 + 2*x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*x_6 - 2*x_6^2,
     x_0*x_4 + x_1*x_3 - x_1*x_5 + x_2*x_4 + 2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_4*x_5 - 2*x_5*x_6 - 2*x_6^2,
     x_0*x_5 -x_1*x_5 - x_2*x_5 - x_2*x_6 + 2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6,
     x_0*x_6 + x_1*x_5 + x_1*x_6 + x_2*x_6 + -4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2*x_4*x_5 + 4*x_5*x_6 + x_6^2
     });

o3 : Ideal of T

i4 : L=flatten entries gens gb(I,DegreeLimit=>6);

i5 : L3=select(L, i -> degree i == {3,0});

i6 : L3c=unique apply(L3, i -> leadCoefficient i);

i7 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
2*c_3*c_11-2*c_6*c_11-2*c_3+2*c_6+12*c_11-12
8*c_3*c_11-8*c_6*c_11+24*c_11^2-8*c_3+8*c_6-24
144*c_3*c_8-144*c_6*c_8-32*c_3*c_9+32*c_6*c_9+432*c_8*c_11-96*c_9*c_11+48*c_3-48*c_6+432*c_8-96*c_9+144*c_11+144
8*c_6*c_8*c_11+12*c_11^3-8*c_6*c_8+8*c_6*c_11+8*c_9*c_11-8*c_6-8*c_9-12*c_11
6*c_11-6
3*c_11^2-3*c_11
6*c_3-6*c_6+36
4*c_11-4
4*c_3-4*c_6+24
2*c_11-2
12*c_8+12
2*c_3-2*c_6+12
12*c_8*c_11+12
24*c_8+12*c_11+12
36*c_11^2+36*c_11-72
24*c_8+24
4*c_3-4*c_6-8*c_9
12
24
8
96*c_8
288*c_8
This suggests setting \( 6 c_3-6 c_6+36 = 0\), or \(c_6 =c_3+6\). We repeat the calculation with this choice: 

i1 : S=QQ[c_3,c_8,c_9,c_11,Degrees=>{0,0,0,0}];

i2 : c_6=c_3+6;

i3 : T=S[x_0..x_6];

i4 : I=ideal({
     x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
     x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
     x_0*x_1 + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*x_5 - x_5*x_6 - x_6^2),
     x_0*x_2 +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x_4*x_6 + x_5^2 + x_5*x_6),
     x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6 + c_11*(x_3^2 - x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_5*x_6 + x_6^2),
     x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6+ c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
     x_0*x_3 + x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6 + x_3^2 + 2*x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*x_6 - 2*x_6^2,
     x_0*x_4 + x_1*x_3 - x_1*x_5 + x_2*x_4 + 2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_4*x_5 - 2*x_5*x_6 - 2*x_6^2,
     x_0*x_5 -x_1*x_5 - x_2*x_5 - x_2*x_6 + 2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6,
     x_0*x_6 + x_1*x_5 + x_1*x_6 + x_2*x_6 + -4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2*x_4*x_5 + 4*x_5*x_6 + x_6^2
     });

o4 : Ideal of T

i5 : L=flatten entries gens gb(I,DegreeLimit=>6);

i6 : L3=select(L, i -> degree i == {3,0});

i7 : L3c=unique apply(L3, i -> leadCoefficient i);

i8 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
2*c_11-2
4*c_8*c_11^2-16*c_8*c_11-6*c_11^2+12*c_8+4*c_11+2
4*c_11-4
c_11-1
12*c_8+12
12*c_8*c_11+12
24*c_8+12*c_11+12
6*c_11^2+6*c_11-12
8*c_9+24
24*c_8+24
12
24
8
32*c_8
96*c_8
This suggests setting \( c_{11} =1\) and \( c_{8} =-1.\) We repeat the calculation with these choices: 

i1 : S=QQ[c_3,c_9,Degrees=>{0,0}];

i2 : c_6=c_3+6;

i3 : c_11=1;

i4 : c_8=-1;

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

i6 : I=ideal({
     x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
     x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
     x_0*x_1 + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*x_5 - x_5*x_6 - x_6^2),
     x_0*x_2 +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x_4*x_6 + x_5^2 + x_5*x_6),
     x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6 + c_11*(x_3^2 - x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_5*x_6 + x_6^2),
     x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6+ c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
     x_0*x_3 + x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6 + x_3^2 + 2*x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*x_6 - 2*x_6^2,
     x_0*x_4 + x_1*x_3 - x_1*x_5 + x_2*x_4 + 2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_4*x_5 - 2*x_5*x_6 - 2*x_6^2,
     x_0*x_5 -x_1*x_5 - x_2*x_5 - x_2*x_6 + 2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6,
     x_0*x_6 + x_1*x_5 + x_1*x_6 + x_2*x_6 + -4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2*x_4*x_5 + 4*x_5*x_6 + x_6^2
     });

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_9+3
4*c_9+12
6*c_9+18
3*c_9+9
8
4
12
This suggests the choice \(c_9 = -3\).

This leads to the following conjectural description of the desired family: Fix any value of \(c_{3}\). Then let \[ c_6 = c_{3} +6, \] \[ c_{8} = -1, \] \[ c_9 = -3, \] and \[ c_{11} = 1. \] In the next section we show that at least two different values of \(c_{3}\) yield equations of a smooth curve with the correct automorphisms.

Checking the equations in Magma

We check that for two different values of \(c_{3}\), 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 \(c_{3}=1\): 

> K:=RationalField();
> c_3:=1;
> c_6:=c_3+6;
> c_8:=-1;
> c_9:=-3;
> c_11:=1;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1\
/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
> x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2\
 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
> x_0*x_1 + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*\
x_5 - x_5*x_6 - x_6^2),
> x_0*x_2 +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x\
_4*x_6 + x_5^2 + x_5*x_6),
> x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6 + c_11*(x_3^2 \
- x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_\
5*x_6 + x_6^2),
> x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6+ c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*\
x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
> x_0*x_3 + x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6 + x_3^2 + 2*x_3*x_\
4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*\
x_6 - 2*x_6^2,
> x_0*x_4 + x_1*x_3 - x_1*x_5 + x_2*x_4 + 2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_\
4*x_5 - 2*x_5*x_6 - 2*x_6^2,
> x_0*x_5 -x_1*x_5 - x_2*x_5 - x_2*x_6 + 2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x\
_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6,
> x_0*x_6 + x_1*x_5 + x_1*x_6 + x_2*x_6 + -4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2\
*x_4*x_5 + 4*x_5*x_6 + x_6^2
> ]);
> 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, -1, -1, 0, 0],
> [0, 0, 0, 0, 1, 0, 0],
> [0, 0, 0, 0, 0, 0, 1],
> [0, 0, 0, 0, 0, 1, 0]
> ]);
> B:=Matrix([
> [-1, 0, 0, 0, 0, 0, 0],
> [0, 1, -1, 0, 0, 0, 0],
> [0, 0, -1, 0, 0, 0, 0],
> [0, 0, 0, 0, 0, 1, 0],
> [0, 0, 0, 1, 1, 0, -1],
> [0, 0, 0, 1, 0, 0, 0],
> [0, 0, 0, 1, 0, 1, -1]
> ]);
> 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, 0, 0, 1, 0],
> [0, 0, 0, -1, -1, -1, 1],
> [0, 0, 0, 1, 0, 0, 0],
> [0, 0, 0, 0, 0, 0, 1]
> ]);
> GL7K:=GeneralLinearGroup(7,K);
> Order(A);    
2
> Order(B);
2
> Order(C);
2
> IdentifyGroup(sub<GL7K | A,B,C>);
<36, 10>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_2
x_1
-x_3
-x_3 + x_4
x_6
x_5
and inverse
-x_0
x_2
x_1
-x_3
-x_3 + x_4
x_6
x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1
-x_1 - x_2
x_4 + x_5 + x_6
x_4
x_3 + x_6
-x_4 - x_6
and inverse
-x_0
x_1
-x_1 - x_2
x_4 + x_5 + x_6
x_4
x_3 + x_6
-x_4 - x_6
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
-x_4 + x_5
-x_4
x_3 - x_4
x_4 + x_6
and inverse
-x_0
-x_1
-x_2
-x_4 + x_5
-x_4
x_3 - x_4
x_4 + x_6
Next, we check the value \(c_{3}=17-\zeta_{15}^4\): 
> K<z_15>:=CyclotomicField(15);
> c_3:=17-z_15^4;
> c_6:=c_3+6;
> c_8:=-1;
> c_9:=-3;
> c_11:=1;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1\
/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
> x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2\
 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
> x_0*x_1 + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*\
x_5 - x_5*x_6 - x_6^2),
> x_0*x_2 +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x\
_4*x_6 + x_5^2 + x_5*x_6),
> x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6 + c_11*(x_3^2 \
- x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_\
5*x_6 + x_6^2),
> x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6+ c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*\
x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
> x_0*x_3 + x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6 + x_3^2 + 2*x_3*x_\
4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*\
x_6 - 2*x_6^2,
> x_0*x_4 + x_1*x_3 - x_1*x_5 + x_2*x_4 + 2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_\
4*x_5 - 2*x_5*x_6 - 2*x_6^2,
> x_0*x_5 -x_1*x_5 - x_2*x_5 - x_2*x_6 + 2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x\
_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6,
> x_0*x_6 + x_1*x_5 + x_1*x_6 + x_2*x_6 + -4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2\
*x_4*x_5 + 4*x_5*x_6 + x_6^2
> ]);
> 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, -1, -1, 0, 0],
> [0, 0, 0, 0, 1, 0, 0],
> [0, 0, 0, 0, 0, 0, 1],
> [0, 0, 0, 0, 0, 1, 0]
> ]);
> B:=Matrix([
> [-1, 0, 0, 0, 0, 0, 0],
> [0, 1, -1, 0, 0, 0, 0],
> [0, 0, -1, 0, 0, 0, 0],
> [0, 0, 0, 0, 0, 1, 0],
> [0, 0, 0, 1, 1, 0, -1],
> [0, 0, 0, 1, 0, 0, 0],
> [0, 0, 0, 1, 0, 1, -1]
> ]);
> 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, 0, 0, 1, 0],
> [0, 0, 0, -1, -1, -1, 1],
> [0, 0, 0, 1, 0, 0, 0],
> [0, 0, 0, 0, 0, 0, 1]
> ]);
> GL7K:=GeneralLinearGroup(7,K);
> IdentifyGroup(sub<GL7K | A,B,C>);
<36, 10>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_2
x_1
-x_3
-x_3 + x_4
x_6
x_5
and inverse
-x_0
x_2
x_1
-x_3
-x_3 + x_4
x_6
x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_1
-x_1 - x_2
x_4 + x_5 + x_6
x_4
x_3 + x_6
-x_4 - x_6
and inverse
-x_0
x_1
-x_1 - x_2
x_4 + x_5 + x_6
x_4
x_3 + x_6
-x_4 - x_6
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
-x_2
-x_4 + x_5
-x_4
x_3 - x_4
x_4 + x_6
and inverse
-x_0
-x_1
-x_2
-x_4 + x_5
-x_4
x_3 - x_4
x_4 + 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 : R=QQ[x_0..x_6]

o1 = R

o1 : PolynomialRing

i2 : c_3=1;

i3 : c_6=c_3+6;

i4 : c_8=-1;

i5 : c_9=-3;

i6 : c_11=1;

i7 : I=ideal({
     x_0^2+c_3*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
     x_1^2 + x_1*x_2 + x_2^2+c_6*(x_3^2 - x_3*x_4 - x_3*x_5 + 1/2*x_3*x_6 + x_4^2 + 1/2*x_4*x_5 + 1/2*x_4*x_6 + x_5^2 + 1/2*x_5*x_6 + x_6^2),
     x_0*x_1 + c_8*(x_1^2 + 2*x_1*x_2) + c_9*(-2*x_3*x_4 - x_3*x_6 + x_4^2 + x_4*x_5 - x_5*x_6 - x_6^2),
     x_0*x_2 +c_8*(-2*x_1*x_2 - x_2^2)+c_9*(x_3^2 - x_3*x_5 + x_3*x_6 - x_4^2 - x_4*x_6 + x_5^2 + x_5*x_6),
     x_1*x_3 + x_1*x_4 + 2*x_1*x_6 + 2*x_2*x_3 - x_2*x_4 + x_2*x_6 + c_11*(x_3^2 - x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 + x_4^2 - x_4*x_5 + 2*x_4*x_6 - 2*x_5^2 - x_5*x_6 + x_6^2),
     x_1*x_5 + 2*x_1*x_6 + 2*x_2*x_5 + x_2*x_6+ c_11*(2*x_3^2 -2*x_3*x_4 - 2*x_3*x_5 + x_3*x_6 + 2*x_4^2 + x_4*x_5 + x_4*x_6 - x_5^2- 2*x_5*x_6 - x_6^2),
     x_0*x_3 + x_1*x_4 - x_1*x_5 - x_2*x_3 + x_2*x_4 - x_2*x_6 + x_3^2 + 2*x_3*x_4 + 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 4*x_4*x_5 - 4*x_4*x_6 - 2*x_5^2 - 4*x_5*x_6 - 2*x_6^2,
     x_0*x_4 + x_1*x_3 - x_1*x_5 + x_2*x_4 + 2*x_3*x_4 - 2*x_3*x_6 - x_4^2 - 4*x_4*x_5 - 2*x_5*x_6 - 2*x_6^2,
     x_0*x_5 -x_1*x_5 - x_2*x_5 - x_2*x_6 + 2*x_3^2 - 2*x_3*x_5 + 2*x_3*x_6 - 2*x_4^2 - 2*x_4*x_6 - x_5^2 - 4*x_5*x_6,
     x_0*x_6 + x_1*x_5 + x_1*x_6 + x_2*x_6 + -4*x_3*x_4 - 2*x_3*x_6 + 2*x_4^2 + 2*x_4*x_5 + 4*x_5*x_6 + x_6^2
     });

o7 : Ideal of R

i8 : betti res I

            0  1  2  3  4 5
o8 = total: 1 10 25 25 10 1
         0: 1  .  .  .  . .
         1: . 10 16  9  . .
         2: .  .  9 16 10 .
         3: .  .  .  .  . 1

o8 : BettiTally
By [Schreyer1986], this Betti table implies that this curve has a \(g_6^2\). We get the same Betti table with \(c_{3} = 17-\zeta_{15}^4\).