A genus 6 Riemann surface with automorphism group (72,15)

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

They list a genus 6 Riemann surface with automorphism group (72,15) 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 (2,4,9).

We use Magma to compute equations of this Riemann surface. The main tools are the Eichler trace formula and Gröbner basis techniques for computing a partial flattening stratification in Macaulay2

Obtaining candidate polynomials in `Magma`

We use some Magma code originally developed by David Swinarski during a visit to the University of Sydney in June/July 2011.

Here is the file autcv10e.txt used below.

The matrix generators \(A\) and \(B\) of the automorphism group are found in Appendix B of [Breuer].

Magma V2.21-7     Fri Apr 29 2016 14:28:45 on dyn-209-2-218-176 [Seed = 
460319620]

+-------------------------------------------------------------------+
|       This copy of Magma has been made available through a        |
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|                                                                   |
|                         Simons Foundation                         |
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| covering U.S. Colleges, Universities, Nonprofit Research entities,|
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+-------------------------------------------------------------------+

Type ? for help.  Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> K<z_72>:=CyclotomicField(72);
> z_3:=z_72^24;
> A:=Matrix([
> [0,0,0,0,1,0],
> [0,0,0,1,0,0],
> [0,0,0,0,0,z_3],
> [0,1,0,0,0,0],
> [1,0,0,0,0,0],
> [0,0,z_3^2,0,0,0]
> ]);
> B:=Matrix([
> [0,0,0,0,0,1],
> [0,0,0,0,-1,0],
> [0,0,0,-1,0,0],
> [0,0,1,0,0,0],
> [0,-1,0,0,0,0],
> [-1,0,0,0,0,0]
> ]);
> MatrixGens,MatrixSKG,Q,C:=RunGivenSKMatrixGenerators(72,6,[A,B,(A*B)^-1]);
Set seed to 0.


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


-----------------------------------------
Class |   1  2  3  4  5  6    7    8    9
Size  |   1  3 18  2 18  6    8    8    8
Order |   1  2  2  3  4  6    9    9    9
-----------------------------------------
p  =  2   1  1  1  4  2  4    8    9    7
p  =  3   1  2  3  1  5  2    4    4    4
-----------------------------------------
X.1   +   1  1  1  1  1  1    1    1    1
X.2   +   1  1 -1  1 -1  1    1    1    1
X.3   +   2  2  0  2  0  2   -1   -1   -1
X.4   +   2  2  0 -1  0 -1   Z1 Z1#2 Z1#4
X.5   +   2  2  0 -1  0 -1 Z1#2 Z1#4   Z1
X.6   +   2  2  0 -1  0 -1 Z1#4   Z1 Z1#2
X.7   +   3 -1 -1  3  1 -1    0    0    0
X.8   +   3 -1  1  3 -1 -1    0    0    0
X.9   +   6 -2  0 -3  0  1    0    0    0


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

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

Z1     = (CyclotomicField(9: Sparse := true)) ! [ RationalField() | 0, 0, 0, 0, 
1, 1 ]


Conjugacy Classes of group G
----------------------------
[1]     Order 1       Length 1      
        Rep [1 0 0 0 0 0]
        [0 1 0 0 0 0]
        [0 0 1 0 0 0]
        [0 0 0 1 0 0]
        [0 0 0 0 1 0]
        [0 0 0 0 0 1]

[2]     Order 2       Length 3      
        Rep [-1  0  0  0  0  0]
        [ 0  1  0  0  0  0]
        [ 0  0 -1  0  0  0]
        [ 0  0  0 -1  0  0]
        [ 0  0  0  0  1  0]
        [ 0  0  0  0  0 -1]

[3]     Order 2       Length 18     
        Rep [       0        0        0        0        1        0]
        [       0        0        0        1        0        0]
        [       0        0        0        0        0 z^12 - 1]
        [       0        1        0        0        0        0]
        [       1        0        0        0        0        0]
        [       0        0    -z^12        0        0        0]

[4]     Order 3       Length 2      
        Rep [   -z^12        0        0        0        0        0]
        [       0    -z^12        0        0        0        0]
        [       0        0    -z^12        0        0        0]
        [       0        0        0 z^12 - 1        0        0]
        [       0        0        0        0 z^12 - 1        0]
        [       0        0        0        0        0 z^12 - 1]

[5]     Order 4       Length 18     
        Rep [ 0  0  0  0  0  1]
        [ 0  0  0  0 -1  0]
        [ 0  0  0 -1  0  0]
        [ 0  0  1  0  0  0]
        [ 0 -1  0  0  0  0]
        [-1  0  0  0  0  0]

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

[7]     Order 9       Length 8      
        Rep [        0         0 -z^12 + 1         0         0         0]
        [     z^12         0         0         0         0         0]
        [        0     -z^12         0         0         0         0]
        [        0         0         0         0         0      z^12]
        [        0         0         0 -z^12 + 1         0         0]
        [        0         0         0         0  z^12 - 1         0]

[8]     Order 9       Length 8      
        Rep [        0         0      z^12         0         0         0]
        [       -1         0         0         0         0         0]
        [        0         1         0         0         0         0]
        [        0         0         0         0         0 -z^12 + 1]
        [        0         0         0        -1         0         0]
        [        0         0         0         0         1         0]

[9]     Order 9       Length 8      
        Rep [        0      z^12         0         0         0         0]
        [        0         0     -z^12         0         0         0]
        [       -1         0         0         0         0         0]
        [        0         0         0         0 -z^12 + 1         0]
        [        0         0         0         0         0  z^12 - 1]
        [        0         0         0        -1         0         0]


Is hyperelliptic?  false
Is cyclic trigonal?  true
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1      =[ 0, 0, 0, 0, 0, 0, 0, 0, 1 ]
H^0(C,K) =[ 0, 0, 0, 0, 0, 0, 0, 0, 1 ]
I_2      =[ 1, 0, 0, 1, 0, 0, 0, 1, 0 ]
S_2      =[ 1, 0, 1, 1, 1, 1, 0, 2, 1 ]
H^0(C,2K)=[ 0, 0, 1, 0, 1, 1, 0, 1, 1 ]
I_3      =[ 0, 0, 0, 1, 1, 0, 1, 2, 3 ]
S_3      =[ 0, 0, 1, 1, 1, 1, 3, 3, 5 ]
H^0(C,3K)=[ 0, 0, 1, 0, 0, 1, 2, 1, 2 ]
I2timesS1=[ 0, 0, 0, 1, 1, 1, 1, 1, 4 ]
Is clearly not generated by quadrics? true
Plane quintic obstruction?  true
Matrix Surface Kernel Generators:
Field K Cyclotomic Field of order 72 and degree 24
[
    [       0        0        0        0        1        0]
    [       0        0        0        1        0        0]
    [       0        0        0        0        0 z^12 - 1]
    [       0        1        0        0        0        0]
    [       1        0        0        0        0        0]
    [       0        0    -z^12        0        0        0],

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

    [        0         0      z^12         0         0         0]
    [       -1         0         0         0         0         0]
    [        0         1         0         0         0         0]
    [        0         0         0         0         0 -z^12 + 1]
    [        0         0         0        -1         0         0]
    [        0         0         0         0         1         0]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 1
Multiplicity 1
[
    x_0*x_3 + x_1*x_4 + x_2*x_5
]
I2 contains a 2-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 1
[
    x_0^2 + (-z^16 + z^4)*x_1^2 - z^20*x_2^2,
    x_3^2 + z^8*x_4^2 + z^16*x_5^2
]
I2 contains a 3-dimensional subspace of CharacterRow 8
Dimension 6
Multiplicity 2
[
    x_0*x_4,
    x_0*x_5,
    x_1*x_3,
    x_1*x_5,
    x_2*x_3,
    x_2*x_4
]
Finding cubics:
I3 contains a 2-dimensional subspace of CharacterRow 4
Dimension 2
Multiplicity 1
[
    x_0*x_1*x_5 + z^16*x_0*x_2*x_4 + (z^20 - z^8)*x_1*x_2*x_3,
    x_0*x_4*x_5 + z^16*x_1*x_3*x_5 + (z^20 - z^8)*x_2*x_3*x_4
]
I3 contains a 2-dimensional subspace of CharacterRow 5
Dimension 2
Multiplicity 1
[
    x_0*x_1*x_5 - z^4*x_0*x_2*x_4 + z^8*x_1*x_2*x_3,
    x_0*x_4*x_5 - z^4*x_1*x_3*x_5 + z^8*x_2*x_3*x_4
]
I3 contains a 3-dimensional subspace of CharacterRow 7
Dimension 9
Multiplicity 3
[
    x_0^3 - x_3^3,
    x_0^2*x_1 - x_4*x_5^2,
    x_0^2*x_2 + z^12*x_4^2*x_5,
    x_0*x_1^2 + (-z^12 + 1)*x_3*x_5^2,
    x_0*x_2^2 + (-z^12 + 1)*x_3*x_4^2,
    x_1^3 - x_4^3,
    x_1^2*x_2 + z^12*x_3^2*x_5,
    x_1*x_2^2 - x_3^2*x_4,
    x_2^3 - x_5^3
]
I3 contains a 6-dimensional subspace of CharacterRow 8
Dimension 9
Multiplicity 3
[
    x_0^3 + x_3^3,
    x_0^2*x_1 + x_4*x_5^2,
    x_0^2*x_2 - z^12*x_4^2*x_5,
    x_0*x_1^2 + (z^12 - 1)*x_3*x_5^2,
    x_0*x_2^2 + (z^12 - 1)*x_3*x_4^2,
    x_1^3 + x_4^3,
    x_1^2*x_2 - z^12*x_3^2*x_5,
    x_1*x_2^2 + x_3^2*x_4,
    x_2^3 + x_5^3
]
I3 contains a 18-dimensional subspace of CharacterRow 9
Dimension 30
Multiplicity 5
[
    x_0^2*x_3,
    x_0^2*x_4,
    x_0^2*x_5,
    x_0*x_1*x_3,
    x_0*x_1*x_4,
    x_0*x_2*x_3,
    x_0*x_2*x_5,
    x_0*x_3^2,
    x_0*x_3*x_4,
    x_0*x_3*x_5,
    x_0*x_4^2,
    x_0*x_5^2,
    x_1^2*x_3,
    x_1^2*x_4,
    x_1^2*x_5,
    x_1*x_2*x_4,
    x_1*x_2*x_5,
    x_1*x_3^2,
    x_1*x_3*x_4,
    x_1*x_4^2,
    x_1*x_4*x_5,
    x_1*x_5^2,
    x_2^2*x_3,
    x_2^2*x_4,
    x_2^2*x_5,
    x_2*x_3^2,
    x_2*x_3*x_5,
    x_2*x_4^2,
    x_2*x_4*x_5,
    x_2*x_5^2
]

The output above shows that this surface is cyclic trigonal.

The quadrics in the canonical ideal cut out the smooth scroll in \(\mathbb{P}^5\). Therefore, we first look for quadrics that yield a smooth surface.

The quadrics come from three isotypical subspaces of \(S_2\).

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

x_0*x_3 + x_1*x_4 + x_2*x_5

The second isotypical subspace, which corresponds to the character \( \chi_4\) in the character table shown above, yields the polynomials

x_0^2 + (-z^16 + z^4)*x_1^2 - z^20*x_2^2,
x_3^2 + z^8*x_4^2 + z^16*x_5^2

The third isotypical subspace corresponds to the character \( \chi_{8}\) in the character table shown above. It is spanned by the monomials \( x_0 x_4, x_0 x_5, x_1 x_3, x_1 x_5, x_2 x_3, x_2 x_4\). The two subspaces \( \operatorname{Span} \{ x_0x_4, x_1x_5,x_2 x_3\} \) and \( \operatorname{Span} \{ x_1x_3, x_2 x_4,x_0 x_5\} \) are preserved. We choose an ordered basis of the second span such that the action of \(G\) is given by the same matrices relative to both bases. This leads to the following three quadrics:

c_1*(x_0*x_4) + c_2*(z_3^2*x_1*x_3),
c_1*(x_1*x_5) + c_2*(z_3^2*x_2*x_4),
c_1*(x_2*x_3) + c_2*(x_0*x_5),

Assume that \(c_1\) is nonzero. Then after dividing by \(c_1\) we may assume \(c_1=1\). 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.) We compute the degree two and three elements in a Gröbner basis in Macaulay2 for the ideal generated by these quadrics.

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

i1 : loadPackage("Cyclotomic")

o1 = Cyclotomic

o1 : Package

i2 : K=cyclotomicField(9);

i3 : z_9=K_0;

i4 : z_3=z_9^3;

i5 : S=K[c_2,Degrees=>{0}]

o5 = S

o5 : PolynomialRing

i6 : T=S[x_0..x_5];

i7 : I=ideal({
     x_0*x_3 + x_1*x_4 + x_2*x_5,
     x_0^2 + z_9^-1*x_1^2 +z_9^-2*x_2^2,
     x_3^2 + z_9*x_4^2 + z_9^2*x_5^2,
     x_0*x_4 + c_2*(z_3^2*x_1*x_3),
     x_1*x_5 + c_2*(z_3^2*x_2*x_4),
     x_2*x_3 + c_2*(x_0*x_5)
     });

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_2^2-ww_9^4
c_2^3-ww_9^3-1
1
c_2+ww_9^2

This suggests setting \(c_2 =-\zeta_9^2\). We check that this yields a smooth surface:


> K<z_9>:=CyclotomicField(9);
> z_3:=z_9^3;
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> Y:=Scheme(P5,[x_0*x_3 + x_1*x_4 + x_2*x_5,
> x_0^2 + z_9^-1*x_1^2 +z_9^-2*x_2^2,
> x_3^2 + z_9*x_4^2 + z_9^2*x_5^2,
> x_0*x_4 -z_9^2*(z_3^2*x_1*x_3),
> x_1*x_5 -z_9^2*(z_3^2*x_2*x_4),
> x_2*x_3 -z_9^2*(x_0*x_5)
> ]);
> Dimension(Y);
2
> IsSingular(Y);
false

Next, we seek cubics in the canonical ideal. The output above shows that the extra cubic generators are from the isotypical subspace of \(S_3\) corresponding to the character \(\chi_{8}\). This isotypical subspace is spanned by nine polynomials which naturally split into three subspaces generated by the \(G\) orbits of \(x_0^3 + x_3^3\), \( x_0^2 x_1 + x_4 x_5^2\) and \( x_0 x_1^2 + \zeta_3 x_3 x_5^2\). We choose bases of these three subspaces so that the \(G\) action is the same on all three ordered bases.


> GL6K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL6K | MatrixGens>;
> FindParallelBases(MatrixG,[C[4][1],C[4][6],C[4][9]],[C[4][2],C[4][5],C[4][7]\
]);
[-z^12*x_0*x_2^2 + x_3*x_4^2]
[      x_0^2*x_1 + x_4*x_5^2]
[ x_1^2*x_2 - z^12*x_3^2*x_5]
> FindParallelBases(MatrixG,[C[4][1],C[4][6],C[4][9]],[C[4][3],C[4][4],C[4][8]\
]);
[     -z^12*x_0*x_1^2 + x_3*x_5^2]
[-z^12*x_1*x_2^2 - z^12*x_3^2*x_4]
[      x_0^2*x_2 - z^12*x_4^2*x_5]

Thus, the extra cubic generators in the canonical ideal are of the form


c_3*(x_0^3+x_3^3) +c_4*(z_3^2*x_0*x_2^2+x_3*x_4^2) + c_5*(x_3*x_5^2+z_3^2*x_0*x_1^2),
c_3*(x_1^3+x_4^3) +c_4*(x_0^2*x_1+x_4*x_5^2) + c_5*(z_3^2*x_1*x_2^2+z_3^2*x_3^2*x_4),
c_3*(x_2^3+x_5^3)+ c_4*(x_1^2*x_2+z_3^2*x_3^2*x_5)+ c_5*(x_0^2*x_2+z_3^2*x_4^2*x_5)

After subtracting multiples of the quadrics \(x_0^2 + \zeta_9^{-1} x_1^2 +\zeta_9^{-2} x_2^2\) and \(x_3^2 + \zeta_9 x_4^2 + \zeta_9^2 x_5^2,\) we may assume \(c_3=0\). Assume that \(c_4\) is nonzero. Then after dividing we may assume that \(c_4=1\). Thus, the extra cubic generators in the canonical ideal are of the form


z_3^2*x_0*x_2^2+x_3*x_4^2 + c_5*(x_3*x_5^2+z_3^2*x_0*x_1^2),
x_0^2*x_1+x_4*x_5^2 + c_5*(z_3^2*x_1*x_2^2+z_3^2*x_3^2*x_4),
x_1^2*x_2+z_3^2*x_3^2*x_5+ c_5*(x_0^2*x_2+z_3^2*x_4^2*x_5)

Again we turn to Macaulay2 to compute part of a flattening stratification to find a coefficient \(c_5\) that yield a smooth curve.

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

i1 : loadPackage("Cyclotomic")

o1 = Cyclotomic

o1 : Package

i2 : K=cyclotomicField(9);

i3 : z_9=K_0;

i4 : z_3=z_9^3;

i5 : S=K[c_5,Degrees=>{0}];

i6 : T=S[x_0..x_5];

i7 : I=ideal({
     x_0*x_3 + x_1*x_4 + x_2*x_5,
     x_0^2 + z_9^-1*x_1^2 +z_9^-2*x_2^2,
     x_3^2 + z_9*x_4^2 + z_9^2*x_5^2,
     x_0*x_4 -z_9^2*(z_3^2*x_1*x_3),
     x_1*x_5 -z_9^2*(z_3^2*x_2*x_4),
     x_2*x_3 -z_9^2*(x_0*x_5),
     z_3^2*x_0*x_2^2+x_3*x_4^2+ c_5*(x_3*x_5^2+z_3^2*x_0*x_1^2),
     x_0^2*x_1+x_4*x_5^2 + c_5*(z_3^2*x_1*x_2^2+z_3^2*x_3^2*x_4),
     x_1^2*x_2+z_3^2*x_3^2*x_5+ c_5*(x_0^2*x_2+z_3^2*x_4^2*x_5)
     });

o7 : Ideal of T

i8 : L=flatten entries gens gb(I);

i9 : L4=select(L, i -> degree i == {4,0});

i10 : L4c=unique apply(L4, i -> leadCoefficient i);

i11 : for i from 0 to #L4c-1 do (print toString(L4c_i) << endl)
c_5^2-ww_9*c_5+ww_9^2
1
c_5^2-ww_9^5
c_5^3+ww_9^3

The output c_5^2-ww_9*c_5+ww_9^2 suggests \( c_5 = -\zeta_9^{-2}\). We check in Magma that this yields a smooth curve with the desired automorphisms.

> K<z_9>:=CyclotomicField(9);
> z_3:=z_9^3;
> c_5:=-z_9^-2;
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> X:=Scheme(P5,[x_0*x_3 + x_1*x_4 + x_2*x_5,
> x_0^2 + z_9^-1*x_1^2 +z_9^-2*x_2^2,
> x_3^2 + z_9*x_4^2 + z_9^2*x_5^2,
> x_0*x_4 -z_9^2*(z_3^2*x_1*x_3),
> x_1*x_5 -z_9^2*(z_3^2*x_2*x_4),
> x_2*x_3 -z_9^2*(x_0*x_5),
> z_3^2*x_0*x_2^2+x_3*x_4^2 + c_5*(x_3*x_5^2+z_3^2*x_0*x_1^2),
> x_0^2*x_1+x_4*x_5^2 + c_5*(z_3^2*x_1*x_2^2+z_3^2*x_3^2*x_4),
> x_1^2*x_2+z_3^2*x_3^2*x_5+ c_5*(x_0^2*x_2+z_3^2*x_4^2*x_5)
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=Matrix([
> [0,0,0,0,1,0],
> [0,0,0,1,0,0],
> [0,0,0,0,0,z_3],
> [0,1,0,0,0,0],
> [1,0,0,0,0,0],
> [0,0,z_3^2,0,0,0]
> ]);
> B:=Matrix([
> [0,0,0,0,0,1],
> [0,0,0,0,-1,0],
> [0,0,0,-1,0,0],
> [0,0,1,0,0,0],
> [0,-1,0,0,0,0],
> [-1,0,0,0,0,0]
> ]);
> Order(A);
2
> Order(B);
4
> Order( (A*B)^-1);
9
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K |A,B>);
<72, 15>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
x_4
x_3
(-z_9^3 - 1)*x_5
x_1
x_0
z_9^3*x_2
and inverse
x_4
x_3
(-z_9^3 - 1)*x_5
x_1
x_0
z_9^3*x_2
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_5
-x_4
x_3
-x_2
-x_1
x_0
and inverse
x_5
-x_4
-x_3
x_2
-x_1
-x_0

Cyclic trigonal equation

We find a change of variables to transform the quadrics into standard scroll equations. From the list of conjugacy classes above, we see that the only order three elements are belong to the fourth conjugacy class, so these must be the cyclic trigonal morphisms. The eigenvalues of the representative shown are \( \zeta_3 \) with multiplicity 3 and \( \zeta_3^2 \) with multiplicity 3. Thus, in the notation of [AchterPries2007] we have \(r = 3, s=3\), so \(d_1 = 4, d_2 = 4\), and the target scroll is given by the \(2 \times 2\) minors of \[ \left[ \begin{array}{rrrrr} x_0 & x_1 & x_3 & x_4 \\ x_1 & x_2 & x_4 & x_5 \end{array} \right]. \] We make a series of transformations to map the quadrics above to these quadrics. First, mapping \[ \begin{array}{rcl} x_1 & \mapsto & \zeta_{18} x_1 \\ x_2 & \mapsto & \zeta_{18}^2x_2 \\ x_4 & \mapsto & \zeta_{18}^{-1}x_4 \\ x_5 & \mapsto & \zeta_{18}^{-2} x_5 \end{array} \] maps the ideal of quadrics above to the ideal generated by \[ \begin{array}{c} x_0 x_3+x_1 x_4+x_2 x_5,\\ x_0^2+x_1^2+x_2^2,\\ x_3^2+x_4^2+x_5^2,\\ x_1 x_3-x_0 x_4,\\ x_2 x_4-x_1 x_5,\\\ x_2 x_3-x_0 x_5. \end{array} \] Next, the map \[ \begin{array}{rcl} x_0 & \mapsto & i(x_0+x_2) \\ x_2 & \mapsto & x_0-x_2 \\ x_3 & \mapsto & i(x_3+x_5) \\ x_5 & \mapsto & x_3-x_5 \end{array} \] yields the ideal \[ \begin{array}{c} x_1 x_4-4 x_0 x_5,\\ x_1^2-4 x_0 x_2,\\ x_4^2-4 x_3 x_5,\\ -x_2 x_4+x_1 x_5,\\ -x_1 x_3+x_0 x_4,\\ x_2 x_3-x_0 x_5 \end{array} \] Finally, scaling \(x_0, x_2,x_3,x_5\) by 1/2 yields the ideal \[ \begin{array}{c} x_1 x_4-x_0 x_5,\\ x_1^2- x_0 x_2,\\ x_4^2- x_3 x_5,\\ -x_2 x_4+x_1 x_5,\\ -x_1 x_3+x_0 x_4,\\ x_2 x_3-x_0 x_5 \end{array} \] as desired. These coordinate transformations map the canonical ideal to the ideal generated by the scroll quadrics and \[ \begin{array}{c} x_0^3+(4 \zeta_6-2) x_0^2 x_2+x_0 x_2^2+\zeta_6 x_3^3+(-2 \zeta_6+4) x_3^2 x_5+\zeta_6 x_3 x_5^2,\\ x_0^2 x_1+(4 \zeta_6-2) x_0 x_1 x_2+x_1 x_2^2+\zeta_6 x_3^2 x_4+(-2 \zeta_6+4) x_3 x_4 x_5+\zeta_6 x_4 x_5^2,\\ x_0^2 x_2+(4 \zeta_6-2) x_0 x_2^2+x_2^3+\zeta_6 x_3^2 x_5+(-2 \zeta_6+4) x_3 x_5^2+\zeta_6 x_5^3 \end{array} \] Finally, scaling \(x_3,x_4,x_5\) by \( \zeta_{18}^{-1}\) yields the cubics \[ \begin{array}{c} x_0^3+(4 \zeta_6-2) x_0^2 x_2+x_0 x_2^2+x_3^3+(-4 \zeta_6+2) x_3^2 x_5+x_3 x_5^2,\\ x_0^2 x_1+(4 \zeta_6-2) x_0 x_1 x_2+x_1 x_2^2+x_3^2 x_4+(-4 \zeta_6+2) x_3 x_4 x_5+x_4 x_5^2\\ x_0^2 x_2+(4 \zeta_6-2) x_0 x_2^2+x_2^3+x_3^2 x_5+(-4 z_6+2) x_3 x_5^2+x_5^3 \end{array} \] These cubics encode the trigonal equation \( y^3 = (x^4-(4\zeta_6-2) x^2+1) (x^4+(4\zeta_6-2) x^2+1)^2\), or \( y^3 = (x^4-2 \sqrt{-3} x^2+1) (x^4+ 2 \sqrt{-3} x^2+1)^2\). This is a cyclic trigonal curve branching over a cube inscribed on the Riemann sphere.

We transform the matrix surface kernel generators using the maps described above and check everything one last time in Magma.

> K<z_36>:=CyclotomicField(36);
> z_12:=z_36^3;
> z_9:=z_36^4;
> z_6:=z_36^6;
> z_3:=z_36^12;
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> X:=Scheme(P5,[
> x_0*x_2-x_1^2,
> x_0*x_4-x_1*x_3,
> x_0*x_5-x_2*x_3,
> x_1*x_4-x_0*x_5,
> x_1*x_5-x_2*x_4,
> x_3*x_5-x_4^2,
> x_0^3+(4*z_6-2)*x_0^2*x_2+x_0*x_2^2+x_3^3+(-4*z_6+2)*x_3^2*x_5+x_3*x_5^2,
> x_0^2*x_1+(4*z_6-2)*x_0*x_1*x_2+x_1*x_2^2+x_3^2*x_4+(-4*z_6+2)*x_3*x_4*x_5+x\
_4*x_5^2,
> x_0^2*x_2+(4*z_6-2)*x_0*x_2^2+x_2^3+x_3^2*x_5+(-4*z_6+2)*x_3*x_5^2+x_5^3
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=Matrix([
> [0,  0,  0,  -z_9/2,  z_36^13/2,  z_9/2],
> [0,  0,  0,  -z_36^13,  0,  -z_36^13],
> [0,  0,  0,  z_9/2,  z_36^13/2,  -z_9/2],
> [z_36^14/2,  z_36^5/2,  -z_36^14/2,  0,  0,  0],
> [-z_36^5,  0,  -z_36^5,  0,  0,  0],
> [-z_36^14/2,  z_36^5/2,  z_36^14/2,  0,  0,  0]
> ]);
> B:=Matrix([
> [0,  0,  0,  z_12^5,  0,  0],
> [0,  0,  0,  0,  z_3^2,  0],
> [0,  0,  0,  0,  0,  -z_12^5],
> [z_12,  0,  0,  0,  0,  0],
> [0,  z_3,  0,  0,  0,  0],
> [0,  0,  -z_12,  0,  0,  0]
> ]);
> Order(A);
2
> Order(B);
4
> Order( (A*B)^(-1));
9
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K | A,B>);
<72, 15>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
1/2*(z_36^8 - z_36^2)*x_3 - z_36^5*x_4 + 1/2*(-z_36^8 + z_36^2)*x_5
1/2*z_36^5*x_3 + 1/2*z_36^5*x_5
1/2*(-z_36^8 + z_36^2)*x_3 - z_36^5*x_4 + 1/2*(z_36^8 - z_36^2)*x_5
-1/2*z_36^4*x_0 + (-z_36^7 + z_36)*x_1 + 1/2*z_36^4*x_2
1/2*(z_36^7 - z_36)*x_0 + 1/2*(z_36^7 - z_36)*x_2
1/2*z_36^4*x_0 + (-z_36^7 + z_36)*x_1 - 1/2*z_36^4*x_2
and inverse
1/2*(z_36^8 - z_36^2)*x_3 - z_36^5*x_4 + 1/2*(-z_36^8 + z_36^2)*x_5
1/2*z_36^5*x_3 + 1/2*z_36^5*x_5
1/2*(-z_36^8 + z_36^2)*x_3 - z_36^5*x_4 + 1/2*(z_36^8 - z_36^2)*x_5
-1/2*z_36^4*x_0 + (-z_36^7 + z_36)*x_1 + 1/2*z_36^4*x_2
1/2*(z_36^7 - z_36)*x_0 + 1/2*(z_36^7 - z_36)*x_2
1/2*z_36^4*x_0 + (-z_36^7 + z_36)*x_1 - 1/2*z_36^4*x_2
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
z_36^3*x_3
(z_36^6 - 1)*x_4
-z_36^3*x_5
(z_36^9 - z_36^3)*x_0
-z_36^6*x_1
(-z_36^9 + z_36^3)*x_2
and inverse
-z_36^3*x_3
(z_36^6 - 1)*x_4
z_36^3*x_5
(-z_36^9 + z_36^3)*x_0
-z_36^6*x_1
(z_36^9 - z_36^3)*x_2

A genus 6 Riemann surface with automorphism group (72,15)

Obtaining candidate polynomials in Magma

Cyclic trigonal equation

Obtaining candidate polynomials in `Magma`