Fordham University

A genus 6 Riemann surface with automorphism group (21,2)

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 (21,2) in the GAP library of small groups. The quotient of this surface by its automorphism group has genus zero, and the quotient morphism is branched over three points with ramification indices (3,7,21).

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

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. 
Magma V2.21-7     Fri May  6 2016 17:06:17 on Davids-MacBook-Pro-2 [Seed = 
2231293111]

+-------------------------------------------------------------------+
|       This copy of Magma has been made available through a        |
|                   generous initiative of the                      |
|                                                                   |
|                         Simons Foundation                         |
|                                                                   |
| covering U.S. Colleges, Universities, Nonprofit Research entities,|
|               and their students, faculty, and staff              |
+-------------------------------------------------------------------+

Type ? for help.  Type -D to quit.
> load "autcv10e.txt";
Loading "autcv10e.txt"
> G:=SmallGroup(21,2);
> MatrixGens,MatrixSKG,Q,C:=RunExample(G,6,[3,7,21]);
Set seed to 0.


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


-------------------------------------------------------------------------------
Class |   1   2   3    4    5    6    7    8    9    10    11    12    13    14
Size  |   1   1   1    1    1    1    1    1    1     1     1     1     1     1
Order |   1   3   3    7    7    7    7    7    7    21    21    21    21    21
-------------------------------------------------------------------------------
p  =  3   1   1   1    9    6    8    5    4    7     5     9     6     8     5
p  =  7   1   2   3    1    1    1    1    1    1     2     3     3     3     3
-------------------------------------------------------------------------------
X.1   +   1   1   1    1    1    1    1    1    1     1     1     1     1     1
X.2   0   1-1-J   J    1    1    1    1    1    1  -1-J     J     J     J     J
X.3   0   1   J-1-J    1    1    1    1    1    1     J  -1-J  -1-J  -1-J  -1-J
X.4   0   1   1   1   Z1 Z1#6 Z1#4 Z1#2 Z1#5 Z1#3  Z1#2    Z1  Z1#6  Z1#4  Z1#2
X.5   0   1-1-J   J   Z1 Z1#6 Z1#4 Z1#2 Z1#5 Z1#3    Z2 Z2#11 Z2#17  Z2#2  Z2#8
X.6   0   1   J-1-J   Z1 Z1#6 Z1#4 Z1#2 Z1#5 Z1#3  Z2#8  Z2#4 Z2#10 Z2#16    Z2
X.7   0   1   1   1 Z1#2 Z1#5   Z1 Z1#4 Z1#3 Z1#6  Z1#4  Z1#2  Z1#5    Z1  Z1#4
X.8   0   1-1-J   J Z1#2 Z1#5   Z1 Z1#4 Z1#3 Z1#6 Z2#16  Z2#8 Z2#20 Z2#11  Z2#2
X.9   0   1   J-1-J Z1#2 Z1#5   Z1 Z1#4 Z1#3 Z1#6  Z2#2    Z2 Z2#13  Z2#4 Z2#16
X.10  0   1   1   1 Z1#3 Z1#4 Z1#5 Z1#6   Z1 Z1#2  Z1#6  Z1#3  Z1#4  Z1#5  Z1#6
X.11  0   1-1-J   J Z1#3 Z1#4 Z1#5 Z1#6   Z1 Z1#2 Z2#10  Z2#5  Z2#2 Z2#20 Z2#17
X.12  0   1   J-1-J Z1#3 Z1#4 Z1#5 Z1#6   Z1 Z1#2 Z2#17 Z2#19 Z2#16 Z2#13 Z2#10
X.13  0   1   1   1 Z1#4 Z1#3 Z1#2   Z1 Z1#6 Z1#5    Z1  Z1#4  Z1#3  Z1#2    Z1
X.14  0   1-1-J   J Z1#4 Z1#3 Z1#2   Z1 Z1#6 Z1#5  Z2#4  Z2#2  Z2#5  Z2#8 Z2#11
X.15  0   1   J-1-J Z1#4 Z1#3 Z1#2   Z1 Z1#6 Z1#5 Z2#11 Z2#16 Z2#19    Z2  Z2#4
X.16  0   1   1   1 Z1#5 Z1#2 Z1#6 Z1#3 Z1#4   Z1  Z1#3  Z1#5  Z1#2  Z1#6  Z1#3
X.17  0   1-1-J   J Z1#5 Z1#2 Z1#6 Z1#3 Z1#4   Z1 Z2#19 Z2#20  Z2#8 Z2#17  Z2#5
X.18  0   1   J-1-J Z1#5 Z1#2 Z1#6 Z1#3 Z1#4   Z1  Z2#5 Z2#13    Z2 Z2#10 Z2#19
X.19  0   1   1   1 Z1#6   Z1 Z1#3 Z1#5 Z1#2 Z1#4  Z1#5  Z1#6    Z1  Z1#3  Z1#5
X.20  0   1-1-J   J Z1#6   Z1 Z1#3 Z1#5 Z1#2 Z1#4 Z2#13 Z2#17 Z2#11  Z2#5 Z2#20
X.21  0   1   J-1-J Z1#6   Z1 Z1#3 Z1#5 Z1#2 Z1#4 Z2#20 Z2#10  Z2#4 Z2#19 Z2#13


---------------------------------------------------
Class |      15    16    17    18    19    20    21
Size  |       1     1     1     1     1     1     1
Order |      21    21    21    21    21    21    21
---------------------------------------------------
p  =  3       4     4     7     9     6     8     7
p  =  7       3     2     2     2     2     2     3
---------------------------------------------------
X.1   +       1     1     1     1     1     1     1
X.2   0       J  -1-J  -1-J  -1-J  -1-J  -1-J     J
X.3   0    -1-J     J     J     J     J     J  -1-J
X.4   0    Z1#5  Z1#5  Z1#3    Z1  Z1#6  Z1#4  Z1#3
X.5   0   Z2#20 Z2#13 Z2#19  Z2#4 Z2#10 Z2#16  Z2#5
X.6   0   Z2#13 Z2#20  Z2#5 Z2#11 Z2#17  Z2#2 Z2#19
X.7   0    Z1#3  Z1#3  Z1#6  Z1#2  Z1#5    Z1  Z1#6
X.8   0    Z2#5 Z2#19 Z2#10    Z2 Z2#13  Z2#4 Z2#17
X.9   0   Z2#19  Z2#5 Z2#17  Z2#8 Z2#20 Z2#11 Z2#10
X.10  0      Z1    Z1  Z1#2  Z1#3  Z1#4  Z1#5  Z1#2
X.11  0   Z2#11  Z2#4    Z2 Z2#19 Z2#16 Z2#13  Z2#8
X.12  0    Z2#4 Z2#11  Z2#8  Z2#5  Z2#2 Z2#20    Z2
X.13  0    Z1#6  Z1#6  Z1#5  Z1#4  Z1#3  Z1#2  Z1#5
X.14  0   Z2#17 Z2#10 Z2#13 Z2#16 Z2#19    Z2 Z2#20
X.15  0   Z2#10 Z2#17 Z2#20  Z2#2  Z2#5  Z2#8 Z2#13
X.16  0    Z1#4  Z1#4    Z1  Z1#5  Z1#2  Z1#6    Z1
X.17  0    Z2#2 Z2#16  Z2#4 Z2#13    Z2 Z2#10 Z2#11
X.18  0   Z2#16  Z2#2 Z2#11 Z2#20  Z2#8 Z2#17  Z2#4
X.19  0    Z1#2  Z1#2  Z1#4  Z1#6    Z1  Z1#3  Z1#4
X.20  0    Z2#8    Z2 Z2#16 Z2#10  Z2#4 Z2#19  Z2#2
X.21  0      Z2  Z2#8  Z2#2 Z2#17 Z2#11  Z2#5 Z2#16


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

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

J = RootOfUnity(3)

Z1     = (CyclotomicField(7: Sparse := true)) ! [ RationalField() | 0, 0, 0, 1, 
0, 0 ]

Z2     = (CyclotomicField(21: Sparse := true)) ! [ RationalField() | 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1 ]


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

[2]     Order 3       Length 1      
        Rep G.1^2

[3]     Order 3       Length 1      
        Rep G.1

[4]     Order 7       Length 1      
        Rep G.2^3

[5]     Order 7       Length 1      
        Rep G.2^4

[6]     Order 7       Length 1      
        Rep G.2^5

[7]     Order 7       Length 1      
        Rep G.2^6

[8]     Order 7       Length 1      
        Rep G.2

[9]     Order 7       Length 1      
        Rep G.2^2

[10]    Order 21      Length 1      
        Rep G.1^2 * G.2^6

[11]    Order 21      Length 1      
        Rep G.1 * G.2^3

[12]    Order 21      Length 1      
        Rep G.1 * G.2^4

[13]    Order 21      Length 1      
        Rep G.1 * G.2^5

[14]    Order 21      Length 1      
        Rep G.1 * G.2^6

[15]    Order 21      Length 1      
        Rep G.1 * G.2

[16]    Order 21      Length 1      
        Rep G.1^2 * G.2

[17]    Order 21      Length 1      
        Rep G.1^2 * G.2^2

[18]    Order 21      Length 1      
        Rep G.1^2 * G.2^3

[19]    Order 21      Length 1      
        Rep G.1^2 * G.2^4

[20]    Order 21      Length 1      
        Rep G.1^2 * G.2^5

[21]    Order 21      Length 1      
        Rep G.1 * G.2^2


Surface kernel generators:  [ G.1^2, G.2, G.1 * G.2^6 ]
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
S_1      =[ 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
H^0(C,K) =[ 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0 ]
I_2      =[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0 ]
S_2      =[ 0, 1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 0, 1, 2, 0 ]
H^0(C,2K)=[ 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0 ]
I_3      =[ 2, 1, 3, 2, 0, 2, 2, 0, 1, 2, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 3 ]
S_3      =[ 3, 2, 4, 3, 1, 3, 3, 0, 2, 4, 1, 2, 3, 2, 2, 4, 3, 3, 4, 3, 4 ]
H^0(C,3K)=[ 1, 1, 1, 1, 1, 1, 1, 0, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1 ]
I2timesS1=[ 3, 1, 5, 3, 0, 3, 2, 0, 1, 1, 0, 0, 0, 1, 1, 1, 2, 3, 2, 2, 5 ]
Is clearly not generated by quadrics? true
Plane quintic obstruction?  true
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 21 and degree 12
[
    [     z^7        0        0        0        0        0]
    [       0 -z^7 - 1        0        0        0        0]
    [       0        0      z^7        0        0        0]
    [       0        0        0 -z^7 - 1        0        0]
    [       0        0        0        0 -z^7 - 1        0]
    [       0        0        0        0        0 -z^7 - 1],

    [z^3 0 0 0 0 0]
    [0 z^3 0 0 0 0]
    [0 0 z^6 0 0 0]
    [0 0 0 z^6 0 0]
    [0 0 0 0 z^9 0]
    [0 0 0 0 0 z^11 - z^9 + z^8 - z^6 + z^4 - z^3 + z - 1]
]
Matrix Surface Kernel Generators:
[
    [-z^7 - 1        0        0        0        0        0]
    [       0      z^7        0        0        0        0]
    [       0        0 -z^7 - 1        0        0        0]
    [       0        0        0      z^7        0        0]
    [       0        0        0        0      z^7        0]
    [       0        0        0        0        0      z^7],

    [z^3 0 0 0 0 0]
    [0 z^3 0 0 0 0]
    [0 0 z^6 0 0 0]
    [0 0 0 z^6 0 0]
    [0 0 0 0 z^9 0]
    [0 0 0 0 0 z^11 - z^9 + z^8 - z^6 + z^4 - z^3 + z - 1],

    [ z^4    0    0    0    0    0]
    [   0 z^11    0    0    0    0]
    [   0    0    z    0    0    0]
    [   0    0    0  z^8    0    0]
    [   0    0    0    0  z^5    0]
    [   0    0    0    0    0  z^2]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 10
Dimension 2
Multiplicity 2
[
    x_0*x_3,
    x_1*x_2
]
I2 contains a 1-dimensional subspace of CharacterRow 13
Dimension 2
Multiplicity 2
[
    x_0*x_4,
    x_2*x_3
]
I2 contains a 1-dimensional subspace of CharacterRow 14
Dimension 2
Multiplicity 2
[
    x_1*x_4,
    x_3^2
]
I2 contains a 1-dimensional subspace of CharacterRow 16
Dimension 2
Multiplicity 2
[
    x_0*x_5,
    x_2*x_4
]
I2 contains a 1-dimensional subspace of CharacterRow 17
Dimension 2
Multiplicity 2
[
    x_1*x_5,
    x_3*x_4
]
I2 contains a 1-dimensional subspace of CharacterRow 20
Dimension 2
Multiplicity 2
[
    x_3*x_5,
    x_4^2
]
Finding cubics:
I3 contains a 2-dimensional subspace of CharacterRow 1
Dimension 3
Multiplicity 3
[
    x_1*x_3*x_5,
    x_1*x_4^2,
    x_3^2*x_4
]
I3 contains a 1-dimensional subspace of CharacterRow 2
Dimension 2
Multiplicity 2
[
    x_0*x_2*x_5,
    x_2^2*x_4
]
I3 contains a 3-dimensional subspace of CharacterRow 3
Dimension 4
Multiplicity 4
[
    x_0*x_3*x_5,
    x_0*x_4^2,
    x_1*x_2*x_5,
    x_2*x_3*x_4
]
I3 contains a 2-dimensional subspace of CharacterRow 4
Dimension 3
Multiplicity 3
[
    x_1*x_4*x_5,
    x_3^2*x_5,
    x_3*x_4^2
]
I3 contains a 2-dimensional subspace of CharacterRow 6
Dimension 3
Multiplicity 3
[
    x_0*x_4*x_5,
    x_2*x_3*x_5,
    x_2*x_4^2
]
I3 contains a 2-dimensional subspace of CharacterRow 7
Dimension 3
Multiplicity 3
[
    x_1*x_5^2,
    x_3*x_4*x_5,
    x_4^3
]
I3 contains a 1-dimensional subspace of CharacterRow 9
Dimension 2
Multiplicity 2
[
    x_0*x_5^2,
    x_2*x_4*x_5
]
I3 contains a 2-dimensional subspace of CharacterRow 10
Dimension 4
Multiplicity 4
[
    x_0^3,
    x_1^3,
    x_3*x_5^2,
    x_4^2*x_5
]
I3 contains a 1-dimensional subspace of CharacterRow 13
Dimension 3
Multiplicity 3
[
    x_0^2*x_2,
    x_1^2*x_3,
    x_4*x_5^2
]
I3 contains a 1-dimensional subspace of CharacterRow 14
Dimension 2
Multiplicity 2
[
    x_0^2*x_3,
    x_0*x_1*x_2
]
I3 contains a 1-dimensional subspace of CharacterRow 15
Dimension 2
Multiplicity 2
[
    x_0*x_1*x_3,
    x_1^2*x_2
]
I3 contains a 2-dimensional subspace of CharacterRow 16
Dimension 4
Multiplicity 4
[
    x_0*x_2^2,
    x_1^2*x_4,
    x_1*x_3^2,
    x_5^3
]
I3 contains a 2-dimensional subspace of CharacterRow 17
Dimension 3
Multiplicity 3
[
    x_0^2*x_4,
    x_0*x_2*x_3,
    x_1*x_2^2
]
I3 contains a 2-dimensional subspace of CharacterRow 18
Dimension 3
Multiplicity 3
[
    x_0*x_1*x_4,
    x_0*x_3^2,
    x_1*x_2*x_3
]
I3 contains a 2-dimensional subspace of CharacterRow 19
Dimension 4
Multiplicity 4
[
    x_1^2*x_5,
    x_1*x_3*x_4,
    x_2^3,
    x_3^3
]
I3 contains a 2-dimensional subspace of CharacterRow 20
Dimension 3
Multiplicity 3
[
    x_0^2*x_5,
    x_0*x_2*x_4,
    x_2^2*x_3
]
I3 contains a 3-dimensional subspace of CharacterRow 21
Dimension 4
Multiplicity 4
[
    x_0*x_1*x_5,
    x_0*x_3*x_4,
    x_1*x_2*x_4,
    x_2*x_3^2
]

This Riemann surface is cyclic trigonal, and hence the quadrics in its canonical ideal cut out a scroll. We begin by finding quadrics that define a smooth surface.

The quadrics in the canonical ideal of a trigonal curve describe a scroll. The output above shows that the quadrics in the canonical ideal may be written in the form 


x_0*x_3+c_1*x_1*x_2,
x_0*x_4+c_2*x_2*x_3,
x_1*x_4+c_3*x_3^2,
x_0*x_5+c_4*x_2*x_4,
x_1*x_5+c_5*x_3*x_4,
x_3*x_5+c_6*x_4^2
We may assume that \(c_1,\ldots,c_6\) are all nonzero. By scaling \(x_0, x_4,x_5\) we may assume that \(c_1 = c_5 = c_6=-1\). (We choose to scale to make these coefficients -1 rather than +1 out of familiarity with equations of scrolls. See the discussion below.)

For generic values of \(c_2,c_3,c_4\), we get a curve, as the following calculation shows: 


> K:=RationalField();
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_2:=17;
> c_3:=23;
> c_4 := -5;
> Y:=Scheme(P5,[
> x_0*x_3-x_1*x_2,
> x_0*x_4+c_2*(x_2*x_3),
> x_1*x_4+c_3*(x_3^2),
> x_0*x_5+c_4*(x_2*x_4),
> x_1*x_5-x_3*x_4,
> x_3*x_5-x_4^2
> ]);
> Dimension(Y);
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.) We compute the leading coefficients of 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 : S=QQ[c_2,c_3,c_4,Degrees=>{0,0,0}];

i2 : T=S[x_0..x_5];

i3 : I=ideal({
     x_0*x_3-x_1*x_2,
     x_0*x_4+c_2*(x_2*x_3),
     x_1*x_4+c_3*(x_3^2),
     x_0*x_5+c_4*(x_2*x_4),
     x_1*x_5-x_3*x_4,
     x_3*x_5-x_4^2
     });

o3 : Ideal of T

i4 : L=flatten entries gens gb(I);

i5 : Lc=unique apply(L, i -> leadCoefficient i);

i6 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
1
c_4
c_3
c_2
c_3+1
c_2-c_4
c_4+1
c_2+1
c_2-c_3
This suggests setting \( c_2 = c_3 = c_4 = -1\). We check that this yields a surface: 

> K:=RationalField();
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_2:=-1;
> c_3:=-1;
> c_4 :=-1;
> Y:=Scheme(P5,[
> x_0*x_3-x_1*x_2,
> x_0*x_4+c_2*(x_2*x_3),
> x_1*x_4+c_3*(x_3^2),
> x_0*x_5+c_4*(x_2*x_4),
> x_1*x_5-x_3*x_4,
> x_3*x_5-x_4^2
> ]);
> Dimension(Y);
2
> IsSingular(Y);
false
Equations of scrolls can be written as minors of deleted catalecticant matrices. Indeed, the quadrics shown above are the \(2 \times 2\) minors of the following matrix: \[ \left[ \begin{array}{rrrrr} x_0 & x_1 & x_3 & x_4 \\ x_2 & x_3 & x_4 & x_5 \end{array} \right] \] Next, we turn to the cubic generators. The output above indicates that the cubics in the canonical ideal that are not generated by quadrics appear in the isotypical subspaces of \(S_3\) with characters \( \chi_{10}\), \( \chi_{13}\), and \(\chi_{16}\).

The isotypical subspace of \(S_3\) corresponding to the character \( \chi_{10}\) has basis \( \{ x_0^3, x_1^3, x_3 x_5^2, x_4^2 x_5 \}\). We have already established that the quadric \( x_3 x_5 - x_4^2 \) is in the ideal. Thus, after subtracting a multiple of \(x_5 (x_3 x_5 - x_4^2)\), we may assume that the cubic generator we seek is of the form \( x_0^3 + c_7 x_1^3 + c_8 x_3 x_5^2 .\)

The isotypical subspace of \(S_3\) corresponding to the character \( \chi_{13}\) has basis \( \{ x_0^2 x_2, x_1^2 x_3, x_4 x_5^2 \}\). Thus, the cubic generator we seek is of the form \( x_0^2 x_2 + c_9 x_1^2 x_3 + c_{10} x_4 x_5^2.\)

The isotypical subspace of \(S_3\) corresponding to the character \( \chi_{16}\) has basis \( \{ x_0 x_2^2, x_1^2 x_4, x_1 x_3^2,x_5^3 \}\). We have already established that the quadric \( x_1 x_4 - x_3^2 \) is in the ideal. Thus, after subtracting a multiple of \(x_1 (x_1 x_4 - x_3^2)\), we may assume that the cubic generator we seek is of the form \( x_0 x_2^2 + c_{11} x_1^2 x_4 + c_{12} x_5^3 .\)

For generic values of \(c_7,c_8,c_9,c_{10},c_{11},c_{12} \), the ideal defined above appears to be zero-dimensional. Here is an example showing this: 

> K:=RationalField();
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_7:=2;
> c_8:=3;
> c_9:=5;
> c_10:=7;
> c_11:=11;
> c_12:=13;
> X:=Scheme(P5,[
> x_0*x_3-x_1*x_2,
> x_0*x_4-x_2*x_3,
> x_1*x_4-x_3^2,
> x_0*x_5-x_2*x_4,
> x_1*x_5-x_3*x_4,
> x_3*x_5-x_4^2,
> x_0^3+c_7*x_1^3+c_8*x_4^2*x_5,
> x_0^2*x_2+c_9*x_1^2*x_3+c_10*x_4*x_5^2,
> x_0*x_2^2+c_11*x_1*x_3^2+c_12*x_5^3
> ]);
> Dimension(X);
0
Therefore, we once again compute a partial flattening stratification to obtain values of \(c_7,\ldots, c_{12}\) that will yield smooth curve. 
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : S=QQ[c_7,c_8,c_9,c_10,c_11,c_12,Degrees=>{0,0,0,0,0,0}];

i2 : T=S[x_0..x_5];

i3 : I=ideal({
     x_0*x_3-x_1*x_2,
     x_0*x_4-x_2*x_3,
     x_1*x_4-x_3^2,
     x_0*x_5-x_2*x_4,
     x_1*x_5-x_3*x_4,
     x_3*x_5-x_4^2,
     x_0^3+c_7*x_1^3+c_8*x_4^2*x_5,
     x_0^2*x_2+c_9*x_1^2*x_3+c_10*x_4*x_5^2,
     x_0*x_2^2+c_11*x_1*x_3^2+c_12*x_5^3
     });

o3 : Ideal of T

i4 : L=flatten entries gens gb(I);

i5 : Lc=unique apply(L, i -> leadCoefficient i);

i6 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
1
c_8*c_9-c_7*c_10-c_8*c_11+c_10*c_11+c_7*c_12-c_9*c_12
c_9-c_11
c_7-c_11
c_7-c_9
c_8*c_9*c_10*c_12-c_7*c_10^2*c_12-c_8*c_10*c_11*c_12+c_10^2*c_11*c_12-c_8*c_9*c_12^2+2*c_7*c_10*c_12^2-c_9*c_10*c_12^2+c_8*c_11*c_12^2-c_10*c_11*c_12^2-c_7*c_12^3+c_9*c_12^3
c_8*c_9*c_12-c_7*c_10*c_12-c_8*c_11*c_12+c_10*c_11*c_12+c_7*c_12^2-c_9*c_12^2
c_8*c_9*c_10-c_7*c_10^2-c_8*c_10*c_11+c_10^2*c_11+c_7*c_10*c_12-c_9*c_10*c_12
c_9*c_12-c_11*c_12
c_7*c_10*c_12-c_10*c_11*c_12-c_7*c_12^2+c_11*c_12^2
c_7*c_12-c_11*c_12
c_7*c_10-c_10*c_11
c_8*c_9*c_10-c_7*c_10^2-c_8*c_10*c_11+c_10^2*c_11-c_8*c_9*c_12+2*c_7*c_10*c_12-c_9*c_10*c_12+c_8*c_11*c_12-c_10*c_11*c_12-c_7*c_12^2+c_9*c_12^2
c_7*c_10-c_10*c_11-c_7*c_12+c_11*c_12
c_8*c_9*c_12^2-c_7*c_10*c_12^2-c_8*c_11*c_12^2+c_10*c_11*c_12^2+c_7*c_12^3-c_9*c_12^3
c_7*c_12^2-c_11*c_12^2
c_10*c_12-c_12^2
c_8*c_12^2-c_12^3
This suggests setting \(c_7 = c_9 = c_{11}\). We repeat the calculation with these choices: 
Macaulay2, version 1.7
with packages: ConwayPolynomials, Elimination, IntegralClosure, LLLBases,
               PrimaryDecomposition, ReesAlgebra, TangentCone

i1 : S=QQ[c_7,c_8,c_10,c_12,Degrees=>{0,0,0,0}];

i2 : c_9=c_7;

i3 : c_11=c_7;

i4 : T=S[x_0..x_5];

i5 : I=ideal({
     x_0*x_3-x_1*x_2,
     x_0*x_4-x_2*x_3,
     x_1*x_4-x_3^2,
     x_0*x_5-x_2*x_4,
     x_1*x_5-x_3*x_4,
     x_3*x_5-x_4^2,
     x_0^3+c_7*x_1^3+c_8*x_4^2*x_5,
     x_0^2*x_2+c_9*x_1^2*x_3+c_10*x_4*x_5^2,
     x_0*x_2^2+c_11*x_1*x_3^2+c_12*x_5^3
     });

o5 : Ideal of T

i6 : L=flatten entries gens gb(I);

i7 : Lc=unique apply(L, i -> leadCoefficient i);

i8 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
1
c_10-c_12
c_8-c_12
c_8-c_10
c_10*c_12-c_12^2
c_8*c_12-c_12^2
c_8*c_12^2-c_12^3
This suggests setting \(c_8 = c_{10} = c_{12}\). Repeating the calculation with these choices does not yield anything new: 
i9 : S=QQ[c_7,c_8,Degrees=>{0,0}];

i10 : c_9=c_7;

i11 : c_10=c_8;

i12 : c_11=c_7;

i13 : c_12=c_8;

i14 : T=S[x_0..x_5];

i15 : I=ideal({
      x_0*x_3-x_1*x_2,
      x_0*x_4-x_2*x_3,
      x_1*x_4-x_3^2,
      x_0*x_5-x_2*x_4,
      x_1*x_5-x_3*x_4,
      x_3*x_5-x_4^2,
      x_0^3+c_7*x_1^3+c_8*x_4^2*x_5,
      x_0^2*x_2+c_9*x_1^2*x_3+c_10*x_4*x_5^2,
      x_0*x_2^2+c_11*x_1*x_3^2+c_12*x_5^3
      });

o15 : Ideal of T

i16 : L=flatten entries gens gb(I);

i17 : Lc=unique apply(L, i -> leadCoefficient i);

i18 : for i from 0 to #Lc-1 do (print toString(Lc_i) << endl)
1
The reason we get nothing new is because it is possible to scale the variables to make \(c_7 = c_8 = 1\) while preserving the quadrics in the ideal. Namely, scaling by \[ \begin{array}{rcl} x_0 & \mapsto & x_0 \\ x_1 & \mapsto & c_7^{\frac{1}{3}} x_1 \\ x_2 & \mapsto & \left(\frac{c_7}{c_8}\right)^{\frac{1}{7}} x_2 \\ x_3 & \mapsto & c_7^{-\frac{1}{3}} \left(\frac{c_7}{c_8}\right)^{\frac{1}{7}} x_3 \\ x_4 & \mapsto & c_7^{-\frac{1}{3}} \left(\frac{c_7}{c_8}\right)^{\frac{2}{7}} x_4 \\ x_5 & \mapsto & c_7^{-\frac{1}{3}} \left(\frac{c_7}{c_8}\right)^{\frac{3}{7}} x_5. \end{array} \] yields the ideal 

      x_0*x_3-x_1*x_2,
      x_0*x_4-x_2*x_3,
      x_1*x_4-x_3^2,
      x_0*x_5-x_2*x_4,
      x_1*x_5-x_3*x_4,
      x_3*x_5-x_4^2,
      x_0^3+x_1^3+x_4^2*x_5,
      x_0^2*x_2+x_1^2*x_3+x_4*x_5^2,
      x_0*x_2^2+x_1*x_3^2+x_5^3
In the next section we show that these equations describe a smooth curve with the correct automorphisms.

Checking the equations in Magma 

> K<z_21>:=CyclotomicField(21);
> z_7:=z_21^3;
> z_3:=z_21^7;
> 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_2,
> x_0*x_4-x_2*x_3,
> x_1*x_4-x_3^2,
> x_0*x_5-x_2*x_4,
> x_1*x_5-x_3*x_4,
> x_3*x_5-x_4^2,
> x_0^3+x_1^3+x_4^2*x_5,
> x_0^2*x_2+x_1^2*x_3+x_4*x_5^2,
> x_0*x_2^2+x_1*x_3^2+x_5^3
> ]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=DiagonalMatrix([z_3^2,z_3,z_3^2,z_3,z_3,z_3]);
> B:=DiagonalMatrix([z_7,z_7,z_7^2,z_7^2,z_7^3,z_7^4]);
> Order(A);
3
> Order(B);
7
> Order( (A*B)^-1);
21
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K | A,B>);
<21, 2>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
(-z_21^7 - 1)*x_0
z_21^7*x_1
(-z_21^7 - 1)*x_2
z_21^7*x_3
z_21^7*x_4
z_21^7*x_5
and inverse
z_21^7*x_0
(-z_21^7 - 1)*x_1
z_21^7*x_2
(-z_21^7 - 1)*x_3
(-z_21^7 - 1)*x_4
(-z_21^7 - 1)*x_5
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
z_21^3*x_0
z_21^3*x_1
z_21^6*x_2
z_21^6*x_3
z_21^9*x_4
(z_21^11 - z_21^9 + z_21^8 - z_21^6 + z_21^4 - z_21^3 + z_21 - 1)*x_5
and inverse
(-z_21^11 - z_21^4)*x_0
(-z_21^11 - z_21^4)*x_1
(-z_21^8 - z_21)*x_2
(-z_21^8 - z_21)*x_3
(z_21^11 - z_21^9 + z_21^8 - z_21^6 + z_21^4 - z_21^3 + z_21 - 1)*x_4
z_21^9*x_5

Cyclic trigonal equation

From the list of conjugacy classes above, we see that the only order three elements belong to the second and third conjugacy classes, so these must be the cyclic trigonal morphisms. The eigenvalues of the representative shown for the second conjugacy class are \( \zeta_3 \) with multiplicity 4 and \( \zeta_3^2 \) with multiplicity 2. Thus, in the notation of [AchterPries2007] we have \(r = 4, s=2\), so \(d_1 = 7, d_2 = 1\), and the target scroll is given by the \(2 \times 2\) minors of \[ \left[ \begin{array}{rrrrr} x_0 & x_2 & x_3 & x_4 \\ x_1 & x_3 & x_4 & x_5 \end{array} \right] \] We see that permuting the variables \(x_1,x_2\) maps the quadrics obtained above to these. This change of variables yields the following equations of the canonical ideal: the \(2 \times 2\) minors of \[ \left[ \begin{array}{rrrrr} x_0 & x_2 & x_3 & x_4 \\ x_1 & x_3 & x_4 & x_5 \end{array} \right] \] and \[ \begin{array}{c} x_0^3+x_2^3+x_4^2 x_5,\\ x_0^2 x_1+x_2^2 x_3+x_4 x_5^2,\\ x_0 x_1^2+x_2 x_3^2+x_5^3 \end{array} \] These cubics encode the trigonal equation \(y^3 + x^7+1=0\). By scaling \(x_0,x_1\) by \(-1\) we obtain the trigonal equation \(y^3 =x^7+1\).