Fordham University

Finding equations of a 1-parameter family of genus 7 Riemann surfaces with 48 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 7 Riemann surfaces with automorphism group (48,41) 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 (2,2,2,4).

We use GAP, Macaulay2 and 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     Fri Nov 13 2015 11:17:02 on ace-math01 [Seed = 1582171282]
Type ? for help.  Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> G:=SmallGroup(48,41);
> RunExample(G,7,[2,2,2,4]);
Set seed to 0.


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


---------------------------------------------------------
Class |   1  2  3  4  5  6  7  8  9   10   11 12 13 14 15
Size  |   1  1  6  6  6  2  2  2  2    3    3  2  4  4  4
Order |   1  2  2  2  2  3  4  4  4    4    4  6 12 12 12
---------------------------------------------------------
p  =  2   1  1  1  1  1  6  2  2  2    2    2  6 12 12 12
p  =  3   1  2  3  4  5  1  7  8  9   11   10  2  8  9  7
---------------------------------------------------------
X.1   +   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 -1
X.3   +   1  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  1
X.5   +   1  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  1
X.7   +   1  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 -1
X.9   +   2  2  0  0  0 -1  2  2  2    0    0 -1 -1 -1 -1
X.10  +   2  2  0  0  0 -1 -2  2 -2    0    0 -1 -1  1  1
X.11  +   2  2  0  0  0 -1  2 -2 -2    0    0 -1  1  1 -1
X.12  +   2  2  0  0  0 -1 -2 -2  2    0    0 -1  1 -1  1
X.13  0   2 -2  0  0  0  2  0  0  0  2*I -2*I -2  0  0  0
X.14  0   2 -2  0  0  0  2  0  0  0 -2*I  2*I -2  0  0  0
X.15  +   4 -4  0  0  0 -2  0  0  0    0    0  2  0  0  0


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

I = RootOfUnity(4)


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

[2]     Order 2       Length 1      
        Rep G.4

[3]     Order 2       Length 6      
        Rep G.1 * G.2 * G.3

[4]     Order 2       Length 6      
        Rep G.1 * G.3 * G.4

[5]     Order 2       Length 6      
        Rep G.1

[6]     Order 3       Length 2      
        Rep G.5

[7]     Order 4       Length 2      
        Rep G.3

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

[9]     Order 4       Length 2      
        Rep G.2

[10]    Order 4       Length 3      
        Rep G.1 * G.2 * G.4

[11]    Order 4       Length 3      
        Rep G.1 * G.2

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

[13]    Order 12      Length 4      
        Rep G.2 * G.3 * G.4 * G.5

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

[15]    Order 12      Length 4      
        Rep G.3 * G.5


Surface kernel generators:  [ G.1 * G.2 * G.3, G.1 * G.5^2, G.1 * G.3 * G.5^2, 
G.1 * G.2 ]
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, 0 ]
S_1      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1 ]
I_2      =[ 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0 ]
S_2      =[ 2, 2, 0, 2, 0, 0, 0, 2, 1, 2, 2, 2, 1, 0, 1 ]
H^0(C,2K)=[ 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1 ]
I_3      =[ 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 5, 3, 7 ]
S_3      =[ 0, 0, 2, 0, 2, 2, 2, 0, 1, 2, 2, 2, 7, 4, 10 ]
H^0(C,3K)=[ 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 3 ]
I2timesS1=[ 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 6, 4, 10 ]
Is clearly not generated by quadrics? false
No subgroup found

RunExample(
    G: GrpPC : G,
    genus: 7,
    E: [ 2, 2, 2, 4 ]
)
FindMatrixGenerators(
    G: GrpPC : G,
    genus: 7,
    T:   Character Table of Group G --------------------------   --...,
    CCL: Conjugacy Classes of group G ---------------------------- [1...,
    M: [ G.1 * G.2 * G.3, G.1 * G.5^2, G.1 * G.3 * G.5^2, G.1 * G.2...
)
In file "autcv10.txt", line 220, column 28:
>>       ags:=ActionGenerators(GModule(T[i]));
                              ^
Runtime error in 'ActionGenerators': Bad argument types
Argument types given: BoolElt

The error "No subgroup found" tells us that Magma has an internal error when finding the matrix generators of representations with character \( \chi_{6},\) \( \chi_{14},\) and/or \( \chi_{15}\).

We obtain these matrices using GAP. First, we check that GAP and Magma use the same polycyclic generators for the group (48,41), by showing that the GAP relations are satisfied by the Magma group, and the Magma relations are satisfied by the GAP group.

Next, in GAP, we obtain matrix generators for the irreducible representations of this group: 


gap> IrreducibleRepresentations(G);
[ Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], 
      [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], 
      [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ -1 ] ], 
      [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ -1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ -1 ] ], 
      [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ -1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], 
      [ [ 1, 0 ], [ 0, 1 ] ], [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], 
  Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, -1 ] ], 
      [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], 
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ], 
      [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], 
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, -1 ] ], 
      [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], 
      [ [ E(3), 0 ], [ 0, E(3)^2 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, E(4) ], [ -E(4), 0 ] ], [ [ 0, -1 ], [ 1, 0 ] ], 
      [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ], 
      [ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, -E(4) ], [ E(4), 0 ] ], [ [ 0, -1 ], [ 1, 0 ] ], 
      [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ], 
      [ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ f1, f2, f3, f4, f5 ]) -> 
    [ [ [ 0, 0, 0, 1 ], [ 0, 0, -1, 0 ], [ 0, -1, 0, 0 ], [ 1, 0, 0, 0 ] ], 
      [ [ 0, -1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ], 
      [ [ E(4), 0, 0, 0 ], [ 0, -E(4), 0, 0 ], [ 0, 0, E(4), 0 ], 
          [ 0, 0, 0, -E(4) ] ], 
      [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ], 
      [ [ E(3), 0, 0, 0 ], [ 0, E(3), 0, 0 ], [ 0, 0, E(3)^2, 0 ], 
          [ 0, 0, 0, E(3)^2 ] ] ] ]

We load these matrices into Magma, compute the characters of the representations they give, and compare them to the order of the characters in the Magma character table. 

> K:=CyclotomicField(12);
> GL7K:=GeneralLinearGroup(7,K);
> rho:=function(G,K,L)
function>     n:=NumberOfRows(Matrix(L[1]));
function>     GLnK:=GeneralLinearGroup(n,K);
function>     L:=[GLnK!Matrix(L[i]): i in [1..#L]];
function>     return hom< G -> GLnK | L>;
function> end function;
> char:=function(CCLR,f)
function>     return [Trace(f(CCLR[i])) : i in [1..#CCLR]];
function> end function;
> LookupCharacter:=function(T,chi)
function>     for i:=1 to #T do
function|for>         if T[i] eq chi then
function|for|if>             return i;
function|for|if>         end if;
function|for>     end for;
function> end function;
> L:=[
> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ 1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ -1 ] ], [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ 1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ -1 ] ], [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], 
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ \
[ 1, 0 ], [ 0, 1 ] ], [ [ z^4, 0 ], [ 0, z^8 ] ] ],   
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ], \
[ [ 1, 0 ], [ 0, 1 ] ], [ [ z^4, 0 ], [ 0, z^8 ] ] ],
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, 1 ] ], [ [ -1, 0 ], [ 0, -1 ] ], \
[ [ 1, 0 ], [ 0, 1 ] ], [ [ z^4, 0 ], [ 0, z^8 ] ] ],
> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ]\
, [ [ 1, 0 ], [ 0, 1 ] ], [ [ z^4, 0 ], [ 0, z^8 ] ] ],
> [ [ [ 0, z^3 ], [ -z^3, 0 ] ], [ [ 0, -1 ], [ 1, 0 ] ], [ [ z^3, 0 ], [ 0, -\
z^3 ] ], [ [ -1, 0 ], [ 0, -1 ] ], 
> [ [ 1, 0 ], [ 0, 1 ] ] ],
> [ [ [ 0, -z^3 ], [ z^3, 0 ] ], [ [ 0, -1 ], [ 1, 0 ] ], [ [ z^3, 0 ], [ 0, -\
z^3 ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ],
> [ [ [ 0, 0, 0, 1 ], [ 0, 0, -1, 0 ], [ 0, -1, 0, 0 ], [ 1, 0, 0, 0 ] ], [ [ \
0, -1, 0, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ], [ [ z^3, 0, \
0, 0 ], [ 0, -z^3, 0, 0 ], [ 0, 0, z^3, 0 ], [ 0, 0, 0, -z^3 ] ],
> [ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, -1 ] ], [ [ \
z^4, 0, 0, 0 ], [ 0, z^4, 0, 0 ], [ 0, 0, z^8, 0 ], [ 0, 0, 0, z^8 ] ] ]
> ];
> G:=SmallGroup(48,41);
> CCLR:=Classes(G);
> CCLR:=[CCLR[i][3]: i in [1..#CCLR]];
> T:=CharacterTable(G);
> D:=[LookupCharacter(T,CharacterRing(G)!char(CCLR,rho(G,K,L[i]))) : i in [1..\
#L]];
> D;
[ 1, 6, 4, 7, 3, 8, 2, 5, 9, 11, 12, 10, 14, 13, 15 ]

This tells us that the second, thirteenth, and fifteenth representations in GAP have characters \( \chi_6\), \(\chi_{14}\), and \( \chi_{15}\) with respect to the Magma character table. Thus, the matrix generators for the representation \( \chi_6 + \chi_{14}+\chi_{15}\) are: 


> MG:=[
> Matrix([
> [-1,0,0,0,0,0,0],
> [0,0,z^3,0,0,0,0],
> [0,-z^3,0,0,0,0,0],
> [0,0,0,0,0,0,1],
> [0,0,0,0,0,-1,0],
> [0,0,0,0,-1,0,0],
> [0,0,0,1,0,0,0]
> ]),
> 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],
> [0,0,0,0,0,0,-1],
> [0,0,0,0,0,1,0]
> ]),
> Matrix([
> [1,0,0,0,0,0,0],
> [0,z^3,0,0,0,0,0],
> [0,0,-z^3,0,0,0,0],
> [0,0,0,z^3,0,0,0],
> [0,0,0,0,-z^3,0,0],
> [0,0,0,0,0,z^3,0],
> [0,0,0,0,0,0,-z^3]
> ]),
> 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]
> ]),
> 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,z^2-1,0,0,0],
> [0,0,0,0,z^2-1,0,0],
> [0,0,0,0,0,-z^2,0],
> [0,0,0,0,0,0,-z^2]
> ])
> ];
> MG:=[GL7K!MG[i]: i in [1..5]];
> rho:=hom< G -> GL7K | MG>;
> A:=rho(G.1 * G.2 * G.3);
> B:=rho(G.1 * G.5^2);
> C:=rho(G.1 * G.3 * G.5^2);
> D:=rho(G.1 * G.2);
> A;
[  -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  z^3    0]
[   0    0    0    0    0    0 -z^3]
[   0    0    0 -z^3    0    0    0]
[   0    0    0    0  z^3    0    0]
> B;
[      -1        0        0        0        0        0        0]
[       0        0      z^3        0        0        0        0]
[       0     -z^3        0        0        0        0        0]
[       0        0        0        0        0        0  z^2 - 1]
[       0        0        0        0        0 -z^2 + 1        0]
[       0        0        0        0      z^2        0        0]
[       0        0        0     -z^2        0        0        0]
> C;
[      -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        0        0        z]
[       0        0        0        0        0        z        0]
[       0        0        0        0 -z^3 + z        0        0]
[       0        0        0 -z^3 + z        0        0        0]
> D;
[ -1   0   0   0   0   0   0]
[  0 z^3   0   0   0   0   0]
[  0   0 z^3   0   0   0   0]
[  0   0   0   0   0   1   0]
[  0   0   0   0   0   0   1]
[  0   0   0  -1   0   0   0]
[  0   0   0   0  -1   0   0]

We use these matrix surface kernel generators to obtain candidate polynomials. 


> MatrixGens,MatrixSKG,Q,C:=RunGivenSKMatrixGenerators(48,7,[A,B,C,D]);        
Set seed to 0.


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


---------------------------------------------------------
Class |   1  2  3  4  5  6  7  8  9   10   11 12 13 14 15
Size  |   1  1  6  6  6  2  2  2  2    3    3  2  4  4  4
Order |   1  2  2  2  2  3  4  4  4    4    4  6 12 12 12
---------------------------------------------------------
p  =  2   1  1  1  1  1  6  2  2  2    2    2  6 12 12 12
p  =  3   1  2  3  4  5  1  7  8  9   11   10  2  9  8  7
---------------------------------------------------------
X.1   +   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 -1
X.3   +   1  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 -1
X.5   +   1  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  1
X.7   +   1  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  1
X.9   +   2  2  0  0  0 -1  2  2  2    0    0 -1 -1 -1 -1
X.10  +   2  2  0  0  0 -1 -2 -2  2    0    0 -1 -1  1  1
X.11  +   2  2  0  0  0 -1  2 -2 -2    0    0 -1  1  1 -1
X.12  +   2  2  0  0  0 -1 -2  2 -2    0    0 -1  1 -1  1
X.13  0   2 -2  0  0  0  2  0  0  0  2*I -2*I -2  0  0  0
X.14  0   2 -2  0  0  0  2  0  0  0 -2*I  2*I -2  0  0  0
X.15  +   4 -4  0  0  0 -2  0  0  0    0    0  2  0  0  0


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

I = RootOfUnity(4)


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

[2]     Order 2       Length 1      
        Rep [ 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]

[3]     Order 2       Length 6      
        Rep [      -1        0        0        0        0        0        0]
        [       0        0     z^12        0        0        0        0]
        [       0    -z^12        0        0        0        0        0]
        [       0        0        0        0        0        0  z^8 - 1]
        [       0        0        0        0        0 -z^8 + 1        0]
        [       0        0        0        0      z^8        0        0]
        [       0        0        0     -z^8        0        0        0]

[4]     Order 2       Length 6      
        Rep [   -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  z^12     0]
        [    0     0     0     0     0     0 -z^12]
        [    0     0     0 -z^12     0     0     0]
        [    0     0     0     0  z^12     0     0]

[5]     Order 2       Length 6      
        Rep [-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 0 0 z^4]
        [0 0 0 0 0 z^4 0]
        [0 0 0 0 -z^12 + z^4 0 0]
        [0 0 0 -z^12 + z^4 0 0 0]

[6]     Order 3       Length 2      
        Rep [      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 - 1       0       0       0]
        [      0       0       0       0 z^8 - 1       0       0]
        [      0       0       0       0       0    -z^8       0]
        [      0       0       0       0       0       0    -z^8]

[7]     Order 4       Length 2      
        Rep [    1     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^12     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^12]

[8]     Order 4       Length 2      
        Rep [    1     0     0     0     0     0     0]
        [    0     0 -z^12     0     0     0     0]
        [    0 -z^12     0     0     0     0     0]
        [    0     0     0     0 -z^12     0     0]
        [    0     0     0 -z^12     0     0     0]
        [    0     0     0     0     0     0 -z^12]
        [    0     0     0     0     0 -z^12     0]

[9]     Order 4       Length 2      
        Rep [ 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  0 -1]
        [ 0  0  0  0  0  1  0]

[10]    Order 4       Length 3      
        Rep [   -1     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     0     0    -1     0]
        [    0     0     0     0     0     0    -1]
        [    0     0     0     1     0     0     0]
        [    0     0     0     0     1     0     0]

[11]    Order 4       Length 3      
        Rep [  -1    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    0    0    1    0]
        [   0    0    0    0    0    0    1]
        [   0    0    0   -1    0    0    0]
        [   0    0    0    0   -1    0    0]

[12]    Order 6       Length 2      
        Rep [       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 + 1        0]
        [       0        0        0        0        0        0 -z^8 + 1]

[13]    Order 12      Length 4      
        Rep [       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 - 1        0        0]
        [       0        0        0 -z^8 + 1        0        0        0]
        [       0        0        0        0        0        0     -z^8]
        [       0        0        0        0        0      z^8        0]

[14]    Order 12      Length 4      
        Rep [1 0 0 0 0 0 0]
        [0 0 z^12 0 0 0 0]
        [0 z^12 0 0 0 0 0]
        [0 0 0 0 -z^4 0 0]
        [0 0 0 -z^4 0 0 0]
        [0 0 0 0 0 0 -z^12 + z^4]
        [0 0 0 0 0 -z^12 + z^4 0]

[15]    Order 12      Length 4      
        Rep [1 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 0 0 0]
        [0 0 0 0 z^4 0 0]
        [0 0 0 0 0 -z^12 + z^4 0]
        [0 0 0 0 0 0 z^12 - z^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, 0 ]
S_1      =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1 ]
I_2      =[ 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0 ]
S_2      =[ 2, 2, 2, 0, 0, 0, 2, 0, 1, 2, 2, 2, 1, 0, 1 ]
H^0(C,2K)=[ 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1 ]
I_3      =[ 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 5, 3, 7 ]
S_3      =[ 0, 0, 0, 2, 2, 2, 0, 2, 1, 2, 2, 2, 7, 4, 10 ]
H^0(C,3K)=[ 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3 ]
I2timesS1=[ 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 6, 4, 10 ]
Is clearly not generated by quadrics? false
Matrix Surface Kernel Generators:
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     0     0  z^12     0]
    [    0     0     0     0     0     0 -z^12]
    [    0     0     0 -z^12     0     0     0]
    [    0     0     0     0  z^12     0     0],

    [      -1        0        0        0        0        0        0]
    [       0        0     z^12        0        0        0        0]
    [       0    -z^12        0        0        0        0        0]
    [       0        0        0        0        0        0  z^8 - 1]
    [       0        0        0        0        0 -z^8 + 1        0]
    [       0        0        0        0      z^8        0        0]
    [       0        0        0     -z^8        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 0 0 z^4]
    [0 0 0 0 0 z^4 0]
    [0 0 0 0 -z^12 + z^4 0 0]
    [0 0 0 -z^12 + z^4 0 0 0],

    [  -1    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    0    0    1    0]
    [   0    0    0    0    0    0    1]
    [   0    0    0   -1    0    0    0]
    [   0    0    0    0   -1    0    0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 2
Multiplicity 2
[
    x_0^2,
    x_3*x_6 - x_4*x_5
]
I2 contains a 1-dimensional subspace of CharacterRow 2
Dimension 2
Multiplicity 2
[
    x_1^2 - x_2^2,
    x_3*x_5 - x_4*x_6
]
I2 contains a 1-dimensional subspace of CharacterRow 3
Dimension 2
Multiplicity 2
[
    x_1*x_2,
    x_3*x_6 + x_4*x_5
]
I2 contains a 1-dimensional subspace of CharacterRow 7
Dimension 2
Multiplicity 2
[
    x_1^2 + x_2^2,
    x_3*x_5 + x_4*x_6
]
I2 contains a 2-dimensional subspace of CharacterRow 10
Dimension 4
Multiplicity 2
[
    x_1*x_3 + x_2*x_4,
    x_1*x_5 + x_2*x_6,
    x_3^2 + x_4^2,
    x_5^2 + x_6^2
]
I2 contains a 2-dimensional subspace of CharacterRow 11
Dimension 4
Multiplicity 2
[
    x_1*x_4 + x_2*x_3,
    x_1*x_6 + x_2*x_5,
    x_3*x_4,
    x_5*x_6
]
I2 contains a 2-dimensional subspace of CharacterRow 12
Dimension 4
Multiplicity 2
[
    x_1*x_3 - x_2*x_4,
    x_1*x_5 - x_2*x_6,
    x_3^2 - x_4^2,
    x_5^2 - x_6^2
]

The first isotypical subspace, which corresponds to the character \( \chi_1\) in the character table shown above, yields a polynomial of the form \[ c_1 x_0^2 +c_2 ( x_3 x_6 - x_4 x_5) \]

Assume that \(c_1\) and \(c_2\) are nonzero. After scaling \(x_0\) and dividing, 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 \[ c_3 (x_1^2 - x_2^2) +c_4( x_3 x_5 - x_4 x_6). \]

Assume that \(c_3\) and \(c_4\) 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 \[ c_5 (x_1 x_2) +c_6( x_3 x_6 + x_4 x_5). \]

Assume that \(c_5\) is nonzero. After dividing by \(c_5\), we may assume that \(c_5= 1.\)

The fourth isotypical subspace, which corresponds to the character \( \chi_7\) in the character table shown above, yields a polynomial of the form \[ c_7 (x_1^2 + x_2^2) +c_8( x_3 x_5 + x_4 x_6). \]

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 \(1 \times 1\), \(2 \times 2\), 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 an ordered bases of the span of the last two polynomials such that the action of \(G\) is given by the same matrices relative to both ordered bases. 


> GL7K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL7K | MatrixGens>;
> FindParallelBases(MatrixG, [Q[5][1],Q[5][2]],[Q[5][3],Q[5][4]]);
[-z^12*x_5^2 - z^12*x_6^2]
[           x_3^2 + x_4^2]
This yields polynomials of the form 

c_9*(x_1*x_3 + x_2*x_4) +c_10*(-z_12^3*x_5^2 - z_12^3*x_6^2),
c_9*(x_1*x_5 + x_2*x_6) +c_10*(x_3^2 + x_4^2),
Assume that \( c_9 \) is nonzero. Then after dividing by \(c_9\), we may assume that \(c_9 = 1\).

The sixth isotypical subspace corresponds to the character \( \chi_{11}\) in the character table shown above. Again we use the FindParallelBases function to find an ordered bases of the span of the last two polynomials such that the action of \(G\) is given by the same matrices relative to the first two polynomials and this ordered bases. 


> FindParallelBases(MatrixG, [Q[6][1],Q[6][2]],[Q[6][3],Q[6][4]]);
[-z^12*x_5*x_6]
[      x_3*x_4]
This yields polynomials of the form 

c_11*(x_1*x_4 + x_2*x_3)+c_12*(-z_12^3*x_5*x_6),
c_11*(x_1*x_6 + x_2*x_5)+c_12*(x_3*x_4),
Assume that \( c_{11} \) is nonzero. Then after dividing by \(c_{11}\), we may assume that \(c_{11} = 1\).

The seventh isotypical subspace corresponds to the character \( \chi_{12}\) in the character table shown above. Again we use the FindParallelBases function to find an ordered bases of the span of the last two polynomials such that the action of \(G\) is given by the same matrices relative to the first two polynomials and this ordered bases. 


> FindParallelBases(MatrixG, [Q[7][1],Q[7][2]],[Q[7][3],Q[7][4]]);
[-z^12*x_5^2 + z^12*x_6^2]
[           x_3^2 - x_4^2]
This yields polynomials of the form 

c_13*(x_1*x_3 - x_2*x_4) + c_14*(-z_12^3*x_5^2 + z_12^3*x_6^2),
c_13*(x_1*x_5 - x_2*x_6) + c_14*(x_3^2 - x_4^2)
Assume that \( c_{13} \) is nonzero. Then after dividing by \(c_{13}\), 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: \[ \begin{array}{l} x_0^2+ x_3 x_6 - x_4 x_5,\\ x_1^2 - x_2^2 + x_3 x_5 - x_4 x_6,\\ x_1 x_2 + c_6 (x_3 x_6 + x_4 x_5),\\ x_1^2 + x_2^2 + c_8 (x_3 x_5 + x_4 x_6),\\ x_1 x_3 + x_2 x_4 +c_{10} (-i x_5^2 - ix_6^2),\\ x_1 x_5 + x_2 x_6 +c_{10} (x_3^2 + x_4^2),\\ x_1 x_4 + x_2 x_3+c_{12} (-i x_5 x_6),\\ x_1 x_6 + x_2 x_5+c_{12} (x_3 x_4),\\ x_1 x_3 - x_2 x_4 + c_{14} (-i x_5^2 + i x_6^2),\\ x_1 x_5 - x_2 x_6 + c_{14} (x_3^2 - x_4^2) \end{array} \] For generic values of \(c_6,c_8, c_{10},c_{12},c_{14}\), the intersection of these 10 quadrics in \(\mathbb{P}^6\) is empty. Here is an example showing this: 

> K<z_12>:=CyclotomicField(12);
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> c_6:=1;
> c_8:=1;
> c_10:=1;
> c_12:=1;
> c_14:=1;
> X:=Scheme(P6,[
> x_0^2+ x_3*x_6 - x_4*x_5,
> x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
> x_1*x_2 + c_6*(x_3*x_6 + x_4*x_5),
> x_1^2 + x_2^2 + c_8*(x_3*x_5 + x_4*x_6),
> x_1*x_3 + x_2*x_4 +c_10*(-z_12^3*x_5^2 - z_12^3*x_6^2),
> x_1*x_5 + x_2*x_6 +c_10*(x_3^2 + x_4^2),
> x_1*x_4 + x_2*x_3+c_12*(-z_12^3*x_5*x_6),
> x_1*x_6 + x_2*x_5+c_12*(x_3*x_4),
> x_1*x_3 - x_2*x_4 + c_14*(-z_12^3*x_5^2 + z_12^3*x_6^2),
> x_1*x_5 - x_2*x_6 + c_14*(x_3^2 - x_4^2)
> ]);
> 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(12);

i3 : z_12=K_0;

i4 : S=K[c_6,c_8,c_10,c_12,c_14,Degrees=>{0,0,0,0,0}]

o4 = S

o4 : PolynomialRing

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

i6 : I=ideal({
     x_0^2+ x_3*x_6 - x_4*x_5,
     x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
     x_1*x_2 + c_6*(x_3*x_6 + x_4*x_5),
     x_1^2 + x_2^2 + c_8*(x_3*x_5 + x_4*x_6),
     x_1*x_3 + x_2*x_4 +c_10*(-z_12^3*x_5^2 - z_12^3*x_6^2),
     x_1*x_5 + x_2*x_6 +c_10*(x_3^2 + x_4^2),
     x_1*x_4 + x_2*x_3+c_12*(-z_12^3*x_5*x_6),
     x_1*x_6 + x_2*x_5+c_12*(x_3*x_4),
     x_1*x_3 - x_2*x_4 + c_14*(-z_12^3*x_5^2 + z_12^3*x_6^2),
     x_1*x_5 - x_2*x_6 + c_14*(x_3^2 - x_4^2)
     });

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_10*c_12+c_12*c_14+2*ww_12^3*c_6
c_10*c_12-c_12*c_14+ww_12^3*c_8-ww_12^3
c_6*c_10-(1/2)*c_8*c_10-c_6*c_14-(1/2)*c_8*c_14+(1/2)*c_10+(1/2)*c_14
c_10^2*c_12+2*c_10^2*c_14-2*c_10*c_14^2-c_12*c_14^2
c_10*c_14-ww_12^3*c_6
c_12*c_14^2+ww_12^3*c_6*c_12+2*ww_12^3*c_6*c_14
c_10*c_14+(1/2)*ww_12^3*c_8+(1/2)*ww_12^3
c_12*c_14^2+(1/2)*ww_12^3*c_8*c_12-ww_12^3*c_8*c_14+(1/2)*ww_12^3*c_12+ww_12^3*c_14
c_6+(1/2)*c_8+1/2
c_12*c_14-ww_12^3*c_8
c_10*c_12-ww_12^3
c_8*c_10-c_14
c_10^2*c_14+(1/2)*ww_12^3*c_10+(1/2)*ww_12^3*c_14
c_6*c_10*c_14-(1/2)*c_10*c_14+(1/2)*c_14^2+ww_12^3*c_6
c_6*c_10^2+(1/2)*c_10^2-(1/2)*c_10*c_14+ww_12^3*c_6
c_14
c_10
c_6
1
c_10*c_12+2*c_10*c_14+c_12*c_14
c_12*c_14
c_10*c_12
This suggests setting \(c_6+(1/2) c_8+1/2 =0\). We repeat the calculation with this choice: 

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

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(12);

i3 : z_12=K_0;

i4 : S=K[c_6,c_10,c_12,c_14,Degrees=>{0,0,0,0}];

i5 : c_8=-2*c_6-1;

i6 : T=S[x_0..x_6];

i7 : I=ideal({
     x_0^2+ x_3*x_6 - x_4*x_5,
     x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
     x_1*x_2 + c_6*(x_3*x_6 + x_4*x_5),
     x_1^2 + x_2^2 + c_8*(x_3*x_5 + x_4*x_6),
     x_1*x_3 + x_2*x_4 +c_10*(-z_12^3*x_5^2 - z_12^3*x_6^2),
     x_1*x_5 + x_2*x_6 +c_10*(x_3^2 + x_4^2),
     x_1*x_4 + x_2*x_3+c_12*(-z_12^3*x_5*x_6),
     x_1*x_6 + x_2*x_5+c_12*(x_3*x_4),
     x_1*x_3 - x_2*x_4 + c_14*(-z_12^3*x_5^2 + z_12^3*x_6^2),
     x_1*x_5 - x_2*x_6 + c_14*(x_3^2 - x_4^2)
     });

o7 : Ideal of T

i8 : L=flatten entries gens gb(I,DegreeLimit=>6);

i9 : L3=select(L, i -> degree i == {3,0});

i10 : L3c=unique apply(L3, i -> leadCoefficient i);

i11 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
c_6*c_10+c_6*c_12+(1/2)*c_10-(1/2)*c_14
c_10*c_12+c_12*c_14+2*ww_12^3*c_6
c_10*c_12-c_12*c_14-2*ww_12^3*c_6-2*ww_12^3
c_10*c_12*c_14+ww_12^3*c_6*c_10+(1/2)*ww_12^3*c_10-(1/2)*ww_12^3*c_14
c_14
c_6*c_10+(1/2)*c_10
c_6*c_10+(1/2)*c_10-(1/2)*c_14
c_6
c_6*c_10+(1/2)*c_10+(1/2)*c_14
c_10^2*c_12+2*c_10^2*c_14-2*c_10*c_14^2-c_12*c_14^2
c_10*c_14-ww_12^3*c_6
c_12*c_14^2+ww_12^3*c_6*c_12+2*ww_12^3*c_6*c_14
c_12*c_14^2-ww_12^3*c_6*c_12+2*ww_12^3*c_6*c_14+2*ww_12^3*c_14
c_12*c_14+2*ww_12^3*c_6+ww_12^3
c_10*c_12-ww_12^3
c_10
1
c_10*c_12+2*c_10*c_14+c_12*c_14
c_12*c_14
c_10*c_12
This suggests setting \( c_{10} c_{12}-\zeta_{12}^3 =0\), or \(c_{12} = i/c_{10}\). To avoid denominators, we replace \( c_{10}\) by \( 1/c_9\), and repeat the calculation with \(c_{12} = ic_9\). 
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(12);

i3 : z_12=K_0;

i4 : S=K[c_6,c_9,c_14,Degrees=>{0,0,0}];

i5 : c_8=-2*c_6-1;

i6 : c_12=z_12^3*c_9;

i7 : T=S[x_0..x_6];

i8 : I=ideal({
     x_0^2+ x_3*x_6 - x_4*x_5,
     x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
     x_1*x_2 + c_6*(x_3*x_6 + x_4*x_5),
     x_1^2 + x_2^2 + c_8*(x_3*x_5 + x_4*x_6),
     c_9*(x_1*x_3 + x_2*x_4)+(-z_12^3*x_5^2 - z_12^3*x_6^2),
     c_9*(x_1*x_5 + x_2*x_6) +(x_3^2 + x_4^2),
     x_1*x_4 + x_2*x_3+c_12*(-z_12^3*x_5*x_6),
     x_1*x_6 + x_2*x_5+c_12*(x_3*x_4),
     x_1*x_3 - x_2*x_4 + c_14*(-z_12^3*x_5^2 + z_12^3*x_6^2),
     x_1*x_5 - x_2*x_6 + c_14*(x_3^2 - x_4^2)
     });

o8 : Ideal of T

i9 : L=flatten entries gens gb(I,DegreeLimit=>6);

i10 : L3=select(L, i -> degree i == {3,0});

i11 : L3c=unique apply(L3, i -> leadCoefficient i);

i12 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
c_6*c_9^2+(1/2)*ww_12^3*c_9*c_14-ww_12^3*c_6-(1/2)*ww_12^3
c_9^2*c_14+2*c_6*c_9+c_9
c_9*c_14+2*c_6+1
c_6*c_9^2-2*ww_12^3*c_6-ww_12^3
c_9
c_6+1/2
c_6*c_9+ww_12^3*c_14
c_6^2-(1/2)*ww_12^3*c_14^2+(1/2)*c_6
c_9^2*c_14+c_9-2*ww_12^3*c_14
This suggests setting \( c_9 c_{14}+2 c_6+1 =0\), or \(c_{14} = \frac{-2c_6-1}{c_{9}}\). We run the calculation again with this choice: 

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

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(12);

i3 : z_12=K_0;

i4 : S=K[c_6,c_9,Degrees=>{0,0}];

i5 : c_8=-2*c_6-1;

i6 : c_12=z_12^3*c_9;

i7 : T=S[x_0..x_6];

i8 : I=ideal({
     x_0^2+ x_3*x_6 - x_4*x_5,
     x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
     x_1*x_2 + c_6*(x_3*x_6 + x_4*x_5),
     x_1^2 + x_2^2 + c_8*(x_3*x_5 + x_4*x_6),
     c_9*(x_1*x_3 + x_2*x_4)+(-z_12^3*x_5^2 - z_12^3*x_6^2),
     c_9*(x_1*x_5 + x_2*x_6) +(x_3^2 + x_4^2),
     x_1*x_4 + x_2*x_3+c_12*(-z_12^3*x_5*x_6),
     x_1*x_6 + x_2*x_5+c_12*(x_3*x_4),
     c_9*(x_1*x_3 - x_2*x_4) + (-2*c_6-1)*(-z_12^3*x_5^2 + z_12^3*x_6^2),
     c_9*(x_1*x_5 - x_2*x_6) + (-2*c_6-1)*(x_3^2 - x_4^2)
     });

o8 : Ideal of T

i9 : L=flatten entries gens gb(I,DegreeLimit=>6);

i10 : L3=select(L, i -> degree i == {3,0});

i11 : L3c=unique apply(L3, i -> leadCoefficient i);

i12 : for i from 0 to #L3c-1 do (print toString(L3c_i) << endl)
c_6*c_9^2-2*ww_12^3*c_6-ww_12^3
c_9
c_6
c_6+1/2
c_6+1
1
This suggests the choice \( c_6 c_9^2-2 i c_6-i = 0\) , or \( c_6 = \frac{i}{c_9^2-2i}. \)

This leads to the following conjectural description of the desired family: Fix any value of \(c_{9}\). Then let \[ c_6 = \frac{i}{c_9^2-2i}, \] \[ c_{8} = -2 c_6-1, \] \[ c_{12}=i c_9, \] and \[ c_{14} = \frac{-(2 c_6+1)}{c_9}. \] Thus the family is \[ \begin{array}{l} x_0^2+ x_3 x_6 - x_4 x_5,\\ x_1^2 - x_2^2 + x_3 x_5 - x_4 x_6,\\ (c_9^2-2 i) x_1 x_2 + i (x_3 x_6 + x_4 x_5),\\ (c_9^2-2 i) (x_1^2 + x_2^2) - c_9^2 (x_3 x_5 + x_4 x_6),\\ c_9 (x_1 x_3 + x_2 x_4)-i x_5^2 - i x_6^2,\\ c_9 (x_1 x_5 + x_2 x_6) +x_3^2 + x_4^2,\\ x_1 x_4 + x_2 x_3+c_9 x_5 x_6,\\ x_1 x_6 + x_2 x_5+i c_9 (x_3 x_4),\\ (c_9^2-2 i) (x_1 x_3 - x_2 x_4) +c_9 (i x_5^2 - i x_6^2),\\ (c_9^2-2 i) (x_1 x_5 - x_2 x_6) -c_9 (x_3^2 - x_4^2) \end{array} \] In the next section we show that at least two different values of \(c_{9}\) yield equations of a smooth curve with the correct automorphisms.

Checking the equations in Magma

We check that for two different values of \(c_{9}\), 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 \(c_{9}=1\): 
> K<z_12>:=CyclotomicField(12);
> i:=z_12^3;
> c_9:=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+ x_3*x_6 - x_4*x_5,
> x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
> (c_9^2-2*i)*x_1*x_2 + i*(x_3*x_6 + x_4*x_5),
> (c_9^2-2*i)*(x_1^2 + x_2^2) - c_9^2*(x_3*x_5 + x_4*x_6),
> c_9*(x_1*x_3 + x_2*x_4)-i*x_5^2 - i*x_6^2,
> c_9*(x_1*x_5 + x_2*x_6) +x_3^2 + x_4^2,
> x_1*x_4 + x_2*x_3+c_9*x_5*x_6,
> x_1*x_6 + x_2*x_5+i*c_9*(x_3*x_4),
> (c_9^2-2*i)*(x_1*x_3 - x_2*x_4) +c_9*(i*x_5^2 - i*x_6^2),
> (c_9^2-2*i)*(x_1*x_5 - x_2*x_6) -c_9*(x_3^2 - x_4^2)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
12*$.1 - 6
2
> A:=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, z_12^3, 0],
> [0, 0, 0, 0, 0, 0, -z_12^3],
> [0, 0, 0, -z_12^3, 0, 0, 0],
> [0, 0, 0, 0, z_12^3, 0, 0]
> ]);
> B:=Matrix([
> [-1, 0, 0, 0, 0, 0, 0],
> [0, 0, z_12^3, 0, 0, 0, 0],
> [0, -z_12^3, 0, 0, 0, 0, 0],
> [0, 0, 0, 0, 0, 0, z_12^2 - 1],
> [0, 0, 0, 0, 0, -z_12^2 + 1, 0],
> [0, 0, 0, 0, z_12^2, 0, 0],
> [0, 0, 0, -z_12^2, 0, 0, 0]
> ]);
> C:=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, 0, 0, z_12],
> [0, 0, 0, 0, 0, z_12, 0],
> [0, 0, 0, 0, -z_12^3 + z_12, 0, 0],
> [0, 0, 0, -z_12^3 + z_12, 0, 0, 0]
> ]);
> GL7K:=GeneralLinearGroup(7,K);
> IdentifyGroup(sub<GL7K | A,B,C>);
<48, 41>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-x_1
x_2
-z_12^3*x_5
z_12^3*x_6
z_12^3*x_3
-z_12^3*x_4
and inverse
-x_0
-x_1
x_2
-z_12^3*x_5
z_12^3*x_6
z_12^3*x_3
-z_12^3*x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-z_12^3*x_2
z_12^3*x_1
-z_12^2*x_6
z_12^2*x_5
(-z_12^2 + 1)*x_4
(z_12^2 - 1)*x_3
and inverse
-x_0
-z_12^3*x_2
z_12^3*x_1
-z_12^2*x_6
z_12^2*x_5
(-z_12^2 + 1)*x_4
(z_12^2 - 1)*x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_2
x_1
(-z_12^3 + z_12)*x_6
(-z_12^3 + z_12)*x_5
z_12*x_4
z_12*x_3
and inverse
-x_0
x_2
x_1
(-z_12^3 + z_12)*x_6
(-z_12^3 + z_12)*x_5
z_12*x_4
z_12*x_3
Next, we check the value \(c_{9}=17+\zeta_{24}^5\): 
> K<z_24>:=CyclotomicField(24);
> z_12:=z_24^2;
> i:=z_12^3;
> c_9:=17+z_24^5;
> P6<x_0,x_1,x_2,x_3,x_4,x_5,x_6>:=ProjectiveSpace(K,6);
> X:=Scheme(P6,[
> x_0^2+ x_3*x_6 - x_4*x_5,
> x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
> (c_9^2-2*i)*x_1*x_2 + i*(x_3*x_6 + x_4*x_5),
> (c_9^2-2*i)*(x_1^2 + x_2^2) - c_9^2*(x_3*x_5 + x_4*x_6),
> c_9*(x_1*x_3 + x_2*x_4)-i*x_5^2 - i*x_6^2,
> c_9*(x_1*x_5 + x_2*x_6) +x_3^2 + x_4^2,
> x_1*x_4 + x_2*x_3+c_9*x_5*x_6,
> x_1*x_6 + x_2*x_5+i*c_9*(x_3*x_4),
> (c_9^2-2*i)*(x_1*x_3 - x_2*x_4) +c_9*(i*x_5^2 - i*x_6^2),
> (c_9^2-2*i)*(x_1*x_5 - x_2*x_6) -c_9*(x_3^2 - x_4^2)
> ]);
> 
> 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 : 
-x_0
-x_1
x_2
-z_24^6*x_5
z_24^6*x_6
z_24^6*x_3
-z_24^6*x_4
and inverse
-x_0
-x_1
x_2
-z_24^6*x_5
z_24^6*x_6
z_24^6*x_3
-z_24^6*x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
-z_24^6*x_2
z_24^6*x_1
-z_24^4*x_6
z_24^4*x_5
(-z_24^4 + 1)*x_4
(z_24^4 - 1)*x_3
and inverse
-x_0
-z_24^6*x_2
z_24^6*x_1
-z_24^4*x_6
z_24^4*x_5
(-z_24^4 + 1)*x_4
(z_24^4 - 1)*x_3
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
-x_0
x_2
x_1
(-z_24^6 + z_24^2)*x_6
(-z_24^6 + z_24^2)*x_5
z_24^2*x_4
z_24^2*x_3
and inverse
-x_0
x_2
x_1
(-z_24^6 + z_24^2)*x_6
(-z_24^6 + z_24^2)*x_5
z_24^2*x_4
z_24^2*x_3

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(12);

i3 : z_12=K_0;

i4 : i=z_12^3;

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

i6 : c_9=1;

i7 : I=ideal {
     x_0^2+ x_3*x_6 - x_4*x_5,
     x_1^2 - x_2^2 + x_3*x_5 - x_4*x_6,
     (c_9^2-2*i)*x_1*x_2 + i*(x_3*x_6 + x_4*x_5),
     (c_9^2-2*i)*(x_1^2 + x_2^2) - c_9^2*(x_3*x_5 + x_4*x_6),
     c_9*(x_1*x_3 + x_2*x_4)-i*x_5^2 - i*x_6^2,
     c_9*(x_1*x_5 + x_2*x_6) +x_3^2 + x_4^2,
     x_1*x_4 + x_2*x_3+c_9*x_5*x_6,
     x_1*x_6 + x_2*x_5+i*c_9*(x_3*x_4),
     (c_9^2-2*i)*(x_1*x_3 - x_2*x_4) +c_9*(i*x_5^2 - i*x_6^2),
     (c_9^2-2*i)*(x_1*x_5 - x_2*x_6) -c_9*(x_3^2 - x_4^2)
     };

o7 : Ideal of S

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
o12 : BettiTally
By [Schreyer1986], this Betti table implies that this curve has a \(g_6^2\). We get the same Betti table with \(c_{9} = 17+\zeta_{24}^5\).