A pencil of genus 6 curves with automorphism group \(A_5\)

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 pencil of genus 6 curves with automorphism group (60,5) in the GAP library of small groups. This is the alternating group \( A_5\). The quotient of any curve in this pencil by its automorphism group has genus zero, and the quotient morphism is branched over four points with ramification indices (2,2,2,3).

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 originally 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     Thu Sep  3 2015 08:45:39 on ace-math01 [Seed = 1257725750]
Type ? for help.  Type -D to quit.
> load "autcv10.txt";
Loading "autcv10.txt"
> MatrixGens,MatrixSKG,Q,C:=RunExample(SmallGroup(60,5),6,[2,2,2,3]);
Set seed to 0.


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


---------------------------
Class |   1  2  3    4    5
Size  |   1 15 20   12   12
Order |   1  2  3    5    5
---------------------------
p  =  2   1  1  3    5    4
p  =  3   1  2  1    5    4
p  =  5   1  2  3    1    1
---------------------------
X.1   +   1  1  1    1    1
X.2   +   3 -1  0   Z1 Z1#2
X.3   +   3 -1  0 Z1#2   Z1
X.4   +   4  0  1   -1   -1
X.5   +   5  1 -1    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(5: Sparse := true)) ! [ RationalField() | 0, 0, -1, -1
]


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

[2]     Order 2       Length 15     
        Rep (1, 2)(3, 4)

[3]     Order 3       Length 20     
        Rep (1, 2, 3)

[4]     Order 5       Length 12     
        Rep (1, 2, 3, 4, 5)

[5]     Order 5       Length 12     
        Rep (1, 3, 4, 5, 2)


Surface kernel generators:  [
    (1, 4)(3, 5),
    (1, 3)(2, 4),
    (2, 4)(3, 5),
    (1, 5, 4)
]
Is hyperelliptic?  false
Is cyclic trigonal?  false
Multiplicities of irreducibles in relevant G-modules:
I_1      =[ 0, 0, 0, 0, 0 ]
S_1      =[ 0, 1, 1, 0, 0 ]
H^0(C,K) =[ 0, 1, 1, 0, 0 ]
I_2      =[ 1, 0, 0, 0, 1 ]
S_2      =[ 2, 0, 0, 1, 3 ]
H^0(C,2K)=[ 1, 0, 0, 1, 2 ]
I_3      =[ 0, 3, 3, 2, 1 ]
S_3      =[ 0, 5, 5, 4, 2 ]
H^0(C,3K)=[ 0, 2, 2, 2, 1 ]
I2timesS1=[ 0, 3, 3, 2, 2 ]
Is clearly not generated by quadrics? false
Plane quintic obstruction?  false
Matrix generators for action on H^0(C,K):
Field K Cyclotomic Field of order 60 and degree 16
[
    [1 -1 0 0 0 0]
    [0 0 1 0 0 0]
    [0 -1 -z^14 + z^6 + z^4 - 1 0 0 0]
    [0 0 0 z^14 - z^6 - z^4 0 z^14 - z^6 - z^4]
    [0 0 0 -z^14 + z^6 + z^4 0 1]
    [0 0 0 0 -1 1],

    [0 -1 -z^14 + z^6 + z^4 - 1 0 0 0]
    [-z^14 + z^6 + z^4 -1 -1 0 0 0]
    [0 z^14 - z^6 - z^4 + 1 1 0 0 0]
    [0 0 0 -z^14 + z^6 + z^4 1 0]
    [0 0 0 z^14 - z^6 - z^4 - 1 z^14 - z^6 - z^4 -1]
    [0 0 0 -1 0 0]
]
Matrix Surface Kernel Generators:
[
    [z^14 - z^6 - z^4 0 1 0 0 0]
    [0 -1 0 0 0 0]
    [z^14 - z^6 - z^4 0 -z^14 + z^6 + z^4 0 0 0]
    [0 0 0 -1 0 0]
    [0 0 0 0 0 -1]
    [0 0 0 0 -1 0],

    [0 -1 0 0 0 0]
    [-1 0 0 0 0 0]
    [1 -1 -1 0 0 0]
    [0 0 0 0 1 0]
    [0 0 0 1 0 0]
    [0 0 0 z^14 - z^6 - z^4 z^14 - z^6 - z^4 -1],

    [z^14 - z^6 - z^4 -z^14 + z^6 + z^4 - 1 -z^14 + z^6 + z^4 - 1 0 0 0]
    [z^14 - z^6 - z^4 -z^14 + z^6 + z^4 - 1 -z^14 + z^6 + z^4 0 0 0]
    [-1 1 0 0 0 0]
    [0 0 0 -1 0 z^14 - z^6 - z^4]
    [0 0 0 0 -1 1]
    [0 0 0 0 0 1],

    [1 -1 -1 0 0 0]
    [1 0 z^14 - z^6 - z^4 0 0 0]
    [-z^14 + z^6 + z^4 -1 -1 0 0 0]
    [0 0 0 z^14 - z^6 - z^4 - 1 z^14 - z^6 - z^4 z^14 - z^6 - z^4]
    [0 0 0 -z^14 + z^6 + z^4 + 1 1 -z^14 + z^6 + z^4]
    [0 0 0 -z^14 + z^6 + z^4 1 -z^14 + z^6 + z^4]
]
Finding quadrics:
I2 contains a 1-dimensional subspace of CharacterRow 1
Dimension 2
Multiplicity 2
[
    x_0^2 + x_0*x_1 + (-z^14 + z^6 + z^4)*x_0*x_2 + x_1^2 + (-z^14 + z^6 + z^4 -
        1)*x_1*x_2 + x_2^2,
    x_3^2 + (z^14 - z^6 - z^4)*x_3*x_4 + (z^14 - z^6 - z^4)*x_3*x_5 + x_4^2 + 
        x_4*x_5 + x_5^2
]
I2 contains a 5-dimensional subspace of CharacterRow 5
Dimension 15
Multiplicity 3
[
    x_0^2 + 1/3*(z^14 - z^6 - z^4 - 2)*x_2^2,
    x_0*x_1 + 1/3*x_2^2,
    x_0*x_2 + 1/3*(-z^14 + z^6 + z^4 + 1)*x_2^2,
    x_0*x_3 + (-z^14 + z^6 + z^4)*x_1*x_5 + (-z^14 + z^6 + z^4 + 1)*x_2*x_5,
    x_0*x_4 + x_1*x_5 + (z^14 - z^6 - z^4 + 1)*x_2*x_3 + x_2*x_4 + (-z^14 + z^6 
        + z^4)*x_2*x_5,
    x_0*x_5 + x_1*x_5 + (-z^14 + z^6 + z^4)*x_2*x_5,
    x_1^2 + 1/3*(-z^14 + z^6 + z^4 - 3)*x_2^2,
    x_1*x_2 + 1/3*(-z^14 + z^6 + z^4 - 2)*x_2^2,
    x_1*x_3 + (z^14 - z^6 - z^4 + 1)*x_1*x_5 + (z^14 - z^6 - z^4 + 1)*x_2*x_4,
    x_1*x_4 - x_2*x_4,
    x_3^2 + 1/5*(3*z^14 - 3*z^6 - 3*z^4 - 6)*x_5^2,
    x_3*x_4 + 1/5*(3*z^14 - 3*z^6 - 3*z^4 - 1)*x_5^2,
    x_3*x_5 + 1/5*(3*z^14 - 3*z^6 - 3*z^4 - 1)*x_5^2,
    x_4^2 - x_5^2,
    x_4*x_5 + 1/5*(2*z^14 - 2*z^6 - 2*z^4 + 1)*x_5^2
      ]

The output above shows that the ideal contains quadrics from two isotypical subspaces of \(S_2\).

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 + x_0*x_1 + (-z^14 + z^6 + z^4)*x_0*x_2 + x_1^2 + (-z^14 + z^6 + z^4 -1)*x_1*x_2 + x_2^2) + c_2(x_3^2 + (z^14 - z^6 - z^4)*x_3*x_4 + (z^14 - z^6 - z^4)*x_3*x_5 + x_4^2 + x_4*x_5 + x_5^2)

We may assume \(c_1\) and \(c_2\) are nonzero and divide by \(c_1\) to obtain a candidate polynomial of the form


(x_0^2 + x_0*x_1 + (-z^14 + z^6 + z^4)*x_0*x_2 + x_1^2 + (-z^14 + z^6 + z^4 -1)*x_1*x_2 + x_2^2) + c_1(x_3^2 + (z^14 - z^6 - z^4)*x_3*x_4 + (z^14 - z^6 - z^4)*x_3*x_5 + x_4^2 + x_4*x_5 + x_5^2)

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 \(3\times 3\) and \(3 \times 3\),. We therefore partition the fifteen polynomials shown in the output into three sets:

those that contain \(x_0, x_1,x_2\) only,
those that contain \(x_3, x_4,x_5\) only, and
those that contain only monomials \(x_i x_j\) where \( i \in \{ 0,1,2\}\) and \( j \in \{3,4,5\}\).

The FindParallelBases command yields bases of these three five-dimensional spaces such that the group action is given by the same matrix on all three bases:


> GL6K:=Parent(MatrixGens[1]);
> MatrixG:=sub<GL6K | MatrixGens>;
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2],Q[2][3],Q[2][7],Q[2][8]],[Q[2][4]\
,Q[2][5],Q[2][6],Q[2][9],Q[2][10]]);
[x_0*x_3 + 1/2*(-z^14 + z^6 + z^4 + 1)*x_0*x_5 + 1/2*(-3*z^14 + 3*z^6 + 3*z^4 + 
    1)*x_1*x_4 + 1/2*(-3*z^14 + 3*z^6 + 3*z^4 + 1)*x_1*x_5 + 1/2*(3*z^14 - 3*z^6
    - 3*z^4 - 1)*x_2*x_4 + 1/2*(-4*z^14 + 4*z^6 + 4*z^4 + 3)*x_2*x_5]
[1/2*(2*z^14 - 2*z^6 - 2*z^4 - 3)*x_0*x_3 + 1/4*(-6*z^14 + 6*z^6 + 6*z^4 + 
    3)*x_0*x_4 + 1/4*(-5*z^14 + 5*z^6 + 5*z^4 + 1)*x_0*x_5 + 1/4*(9*z^14 - 9*z^6
    - 9*z^4 - 6)*x_1*x_3 + 1/4*(-4*z^14 + 4*z^6 + 4*z^4 + 3)*x_1*x_4 + 
    1/4*(-7*z^14 + 7*z^6 + 7*z^4 + 3)*x_1*x_5 + 1/4*(3*z^14 - 3*z^6 - 3*z^4 - 
    3)*x_2*x_3 + 1/4*(-8*z^14 + 8*z^6 + 8*z^4 + 3)*x_2*x_4 + 1/4*(-z^14 + z^6 + 
    z^4 + 1)*x_2*x_5]
[1/4*(-z^14 + z^6 + z^4 - 1)*x_0*x_3 + 1/4*(3*z^14 - 3*z^6 - 3*z^4 - 3)*x_0*x_4 
    + 1/2*(-z^14 + z^6 + z^4)*x_0*x_5 + 1/4*(z^14 - z^6 - z^4 - 2)*x_1*x_4 + 
    1/4*(z^14 - z^6 - z^4 - 2)*x_1*x_5 + 1/4*(-3*z^14 + 3*z^6 + 3*z^4)*x_2*x_3 +
    1/4*(2*z^14 - 2*z^6 - 2*z^4 - 1)*x_2*x_4 + 1/4*(3*z^14 - 3*z^6 - 3*z^4 - 
    1)*x_2*x_5]
[1/2*(2*z^14 - 2*z^6 - 2*z^4 + 1)*x_0*x_3 + 1/2*(-3*z^14 + 3*z^6 + 3*z^4 + 
    1)*x_0*x_5 + 1/2*(-3*z^14 + 3*z^6 + 3*z^4 + 3)*x_1*x_3 + 1/2*(z^14 - z^6 - 
    z^4 - 1)*x_1*x_4 + 1/2*(z^14 - z^6 - z^4 - 1)*x_1*x_5 + 1/2*(2*z^14 - 2*z^6 
    - 2*z^4 + 1)*x_2*x_4 + 1/2*(-z^14 + z^6 + z^4 + 2)*x_2*x_5]
[1/4*(-z^14 + z^6 + z^4 + 2)*x_0*x_3 + 1/4*(4*z^14 - 4*z^6 - 4*z^4 - 3)*x_0*x_5 
    + 1/4*(-3*z^14 + 3*z^6 + 3*z^4 + 3)*x_1*x_3 + 1/4*(z^14 - z^6 - z^4 + 
    1)*x_1*x_4 + 1/2*(2*z^14 - 2*z^6 - 2*z^4 - 1)*x_1*x_5 + 1/4*(2*z^14 - 2*z^6 
    - 2*z^4 - 1)*x_2*x_4 + 1/4*(3*z^14 - 3*z^6 - 3*z^4 - 1)*x_2*x_5]
> FindParallelBases(MatrixG,[Q[2][1],Q[2][2],Q[2][3],Q[2][7],Q[2][8]],[Q[2][11\
],Q[2][12],Q[2][13],Q[2][14],Q[2][15]]);
[x_3^2 + (-z^14 + z^6 + z^4 - 3)*x_3*x_4 + (3*z^14 - 3*z^6 - 3*z^4 + 3)*x_3*x_5 
    + 1/2*(-3*z^14 + 3*z^6 + 3*z^4 - 4)*x_4^2 + (-3*z^14 + 3*z^6 + 3*z^4 - 
    4)*x_4*x_5 + 1/2*(-z^14 + z^6 + z^4)*x_5^2]
[1/4*(z^14 - z^6 - z^4)*x_3^2 + (z^14 - z^6 - z^4 + 2)*x_3*x_4 + 1/2*(-6*z^14 + 
    6*z^6 + 6*z^4 - 9)*x_3*x_5 + 1/4*(-2*z^14 + 2*z^6 + 2*z^4 - 3)*x_4^2 + 
    1/2*(-2*z^14 + 2*z^6 + 2*z^4 - 3)*x_4*x_5 + 1/4*(-z^14 + z^6 + z^4 - 
    2)*x_5^2]
[1/2*(z^14 - z^6 - z^4 + 1)*x_3^2 + 1/2*(3*z^14 - 3*z^6 - 3*z^4 + 5)*x_3*x_4 + 
    1/2*(-3*z^14 + 3*z^6 + 3*z^4 - 3)*x_3*x_5 + 1/4*(-z^14 + z^6 + z^4 - 
    1)*x_4^2 + 1/2*(-4*z^14 + 4*z^6 + 4*z^4 - 7)*x_4*x_5 + 1/4*(-3*z^14 + 3*z^6 
    + 3*z^4 - 7)*x_5^2]
[1/2*(-4*z^14 + 4*z^6 + 4*z^4 - 5)*x_3^2 + (2*z^14 - 2*z^6 - 2*z^4 + 2)*x_3*x_4 
    + (6*z^14 - 6*z^6 - 6*z^4 + 9)*x_3*x_5 + (z^14 - z^6 - z^4 + 2)*x_4^2 + 
    (2*z^14 - 2*z^6 - 2*z^4 + 4)*x_4*x_5 + 1/2*(5*z^14 - 5*z^6 - 5*z^4 + 
    8)*x_5^2]
[1/4*(-z^14 + z^6 + z^4 - 1)*x_3^2 + 1/2*(3*z^14 - 3*z^6 - 3*z^4 + 5)*x_3*x_4 + 
    1/2*(6*z^14 - 6*z^6 - 6*z^4 + 9)*x_3*x_5 + 1/2*(z^14 - z^6 - z^4 + 1)*x_4^2 
    + 1/2*(5*z^14 - 5*z^6 - 5*z^4 + 8)*x_4*x_5 + 1/4*(6*z^14 - 6*z^6 - 6*z^4 + 
    11)*x_5^2]

For convenience, let \( f_1,\ldots,f_5\) be the Magma polynomials [Q[2][1],Q[2][2],Q[2][3],Q[2][7],Q[2][8]], let \(f_6,\ldots,f_{10}\) be the output of the first FindParallelBases command, and let \(f_{11},\ldots, f_{12}\) be the output of the second FindParallelBases command. Then the candidate polynomials from this isotypical space are of the form \[ \begin{array}{c} c_3 f_1 + c_4 f_6 + c_5 f_{11}\\ c_3 f_2 + c_4 f_7 + c_5 f_{12}\\ c_3 f_3 + c_4 f_8 + c_5 f_{13}\\ c_3 f_4 + c_4 f_9 + c_5 f_{14}\\ c_3 f_5 + c_4 f_{10} + c_5 f_{15} \end{array} \] We may assume that \(c_3,c_4,c_5\) are all nonzero, then scale \(x_3,x_4,x_5\) to make \( c_3 = c_4\), then divide by \(c_3\) to obtain candidate polynomials: \[ \begin{array}{c} f_1 + f_6 + c_5 f_{11}\\ f_2 + f_7 + c_5 f_{12}\\ f_3 + f_8 + c_5 f_{13}\\ f_4 + f_9 + c_5 f_{14}\\ f_5 + f_{10} + c_5 f_{15} \end{array} \] We have thus obtained 6 quadrics with two unknown coefficients as our candidate polynomials. They are shown in the code below. For generic values of c_2 and c_5, the intersection of these 6 quadrics in \(\mathbb{P}^5\) is empty. Here is an example showing this:

> K<z_60>:=CyclotomicField(60);
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_2:=1;
> c_5:=1;
> X:=Scheme(P5,[x_0^2 + x_0*x_1 + (-z_60^14 + z_60^6 + z_60^4)*x_0*x_2 + x_1^2\
 + (-z_60^14 + z_60^6 + z_60^4 - 1)*x_1*x_2 + x_2^2+c_2*(x_3^2 + (z_60^14 - z_\
60^6 - z_60^4)*x_3*x_4 + (z_60^14 - z_60^6 - z_60^4)*x_3*x_5 + x_4^2 + x_4*x_5\
 + x_5^2),
> x_0^2 + 1/3*(z_60^14 - z_60^6 - z_60^4 - 2)*x_2^2 +(x_0*x_3 + 1/2*(-z_60^14 \
+ z_60^6 + z_60^4 + 1)*x_0*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_\
1*x_4 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_1*x_5 + 1/2*(3*z_60^14 - \
3*z_60^6 - 3*z_60^4 - 1)*x_2*x_4 + 1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 + 3)*\
x_2*x_5) +c_5*(x_3^2 + (-z_60^14 + z_60^6 + z_60^4 - 3)*x_3*x_4 + (3*z_60^14 -\
 3*z_60^6 - 3*z_60^4 + 3)*x_3*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 4)\
*x_4^2 + (-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 4)*x_4*x_5 + 1/2*(-z_60^14 + z_60\
^6 + z_60^4)*x_5^2),
> x_0*x_1 + 1/3*x_2^2 +(1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 3)*x_0*x_3 + 1/\
4*(-6*z_60^14 + 6*z_60^6 + 6*z_60^4 + 3)*x_0*x_4 + 1/4*(-5*z_60^14 + 5*z_60^6 \
+ 5*z_60^4 + 1)*x_0*x_5 + 1/4*(9*z_60^14 - 9*z_60^6- 9*z_60^4 - 6)*x_1*x_3 + 1\
/4*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 + 3)*x_1*x_4 + 1/4*(-7*z_60^14 + 7*z_60^6\
 + 7*z_60^4 + 3)*x_1*x_5 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 3)*x_2*x_3 +\
 1/4*(-8*z_60^14 + 8*z_60^6 + 8*z_60^4 + 3)*x_2*x_4 + 1/4*(-z_60^14 + z_60^6 +\
  z_60^4 + 1)*x_2*x_5) +c_5*(1/4*(z_60^14 - z_60^6 - z_60^4)*x_3^2 + (z_60^14 \
- z_60^6 - z_60^4 + 2)*x_3*x_4 + 1/2*(-6*z_60^14 + 6*z_60^6 + 6*z_60^4 - 9)*x_\
3*x_5 + 1/4*(-2*z_60^14 + 2*z_60^6 + 2*z_60^4 - 3)*x_4^2 + 1/2*(-2*z_60^14 + 2\
*z_60^6 + 2*z_60^4 - 3)*x_4*x_5 + 1/4*(-z_60^14 + z_60^6 + z_60^4 - 2)*x_5^2),
> x_0*x_2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 + 1)*x_2^2 +(1/4*(-z_60^14 + z_60^\
6 + z_60^4 - 1)*x_0*x_3 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 3)*x_0*x_4 + \
1/2*(-z_60^14 + z_60^6 + z_60^4)*x_0*x_5 + 1/4*(z_60^14 - z_60^6 - z_60^4 - 2)\
*x_1*x_4 + 1/4*(z_60^14 - z_60^6 - z_60^4 - 2)*x_1*x_5 + 1/4*(-3*z_60^14 + 3*z\
_60^6 + 3*z_60^4)*x_2*x_3 +1/4*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_2*x_4 +\
 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 1)*x_2*x_5) +c_5*(1/2*(z_60^14 - z_60^\
6 - z_60^4 + 1)*x_3^2 + 1/2*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 + 5)*x_3*x_4 + 1/\
2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 3)*x_3*x_5 + 1/4*(-z_60^14 + z_60^6 + z_\
60^4 - 1)*x_4^2 + 1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 - 7)*x_4*x_5 + 1/4*(-3\
*z_60^14 + 3*z_60^6 + 3*z_60^4 - 7)*x_5^2),
> x_1^2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 - 3)*x_2^2 +(1/2*(2*z_60^14 - 2*z_60\
^6 - 2*z_60^4 + 1)*x_0*x_3 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_0*x_\
5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 3)*x_1*x_3 + 1/2*(z_60^14 - z_60^6\
 - z_60^4 - 1)*x_1*x_4 + 1/2*(z_60^14 - z_60^6 - z_60^4 - 1)*x_1*x_5 + 1/2*(2*\
z_60^14 - 2*z_60^6 - 2*z_60^4 + 1)*x_2*x_4 + 1/2*(-z_60^14 + z_60^6 + z_60^4 +\
 2)*x_2*x_5) +c_5*(1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 - 5)*x_3^2 + (2*z_60^\
14 - 2*z_60^6 - 2*z_60^4 + 2)*x_3*x_4 + (6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 9)*\
x_3*x_5 + (z_60^14 - z_60^6 - z_60^4 + 2)*x_4^2 + (2*z_60^14 - 2*z_60^6 - 2*z_\
60^4 + 4)*x_4*x_5 + 1/2*(5*z_60^14 - 5*z_60^6 - 5*z_60^4 + 8)*x_5^2),
> x_1*x_2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 - 2)*x_2^2 +(1/4*(-z_60^14 + z_60^\
6 + z_60^4 + 2)*x_0*x_3 + 1/4*(4*z_60^14 - 4*z_60^6 - 4*z_60^4 - 3)*x_0*x_5 + \
1/4*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 3)*x_1*x_3 + 1/4*(z_60^14 - z_60^6 - z\
_60^4 + 1)*x_1*x_4 + 1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_1*x_5 + 1/4*(\
2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_2*x_4 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z\
_60^4 - 1)*x_2*x_5) +c_5*(1/4*(-z_60^14 + z_60^6 + z_60^4 - 1)*x_3^2 + 1/2*(3*\
z_60^14 - 3*z_60^6 - 3*z_60^4 + 5)*x_3*x_4 + 1/2*(6*z_60^14 - 6*z_60^6 - 6*z_6\
0^4 + 9)*x_3*x_5 + 1/2*(z_60^14 - z_60^6 - z_60^4 + 1)*x_4^2 + 1/2*(5*z_60^14 \
- 5*z_60^6 - 5*z_60^4 + 8)*x_4*x_5 + 1/4*(6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 11\
)*x_5^2)]);
> Dimension(X);
-1

Therefore, we turn to Macaulay2 to compute part of a flattening stratification. (The reason we switch to Macaulay2 is that Magma will not allow us to compute Gröbner bases in a ring of the form \(K[c_2,c_5][x_0,\ldots,x_5]\).)

Partial flattening stratification in `Macaulay2`

We begin to compute a Gröbner basis in Macaulay2 for the ideal generated by the candidate polynomials and then study the first degree three Gröbner basis element:

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

i1 : loadPackage("Cyclotomic");

i2 : K=cyclotomicField(60);

i3 : z_60=K_0;

i4 : R=K[c_2,c_5,Degrees=>{0,0}];

i5 : S=R[x_0..x_5];

i6 : I=ideal {x_0^2 + x_0*x_1 + (-z_60^14 + z_60^6 + z_60^4)*x_0*x_2 + x_1^2 + (-z_60^14 + z_60^6 + z_60^4 - 1)*x_1*x_2 + x_2^2+c_2*(x_3^2 + (z_60^14 - z_60^6 - z_60^4)*x_3*x_4 + (z_60^14 - z_60^6 - z_60^4)*x_3*x_5 + x_4^2 + x_4*x_5 + x_5^2),
     x_0^2 + 1/3*(z_60^14 - z_60^6 - z_60^4 - 2)*x_2^2 +(x_0*x_3 + 1/2*(-z_60^14 + z_60^6 + z_60^4 + 1)*x_0*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_1*x_4 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_1*x_5 + 1/2*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 1)*x_2*x_4 + 1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 + 3)*x_2*x_5) +c_5*(x_3^2 + (-z_60^14 + z_60^6 + z_60^4 - 3)*x_3*x_4 + (3*z_60^14 - 3*z_60^6 - 3*z_60^4 + 3)*x_3*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 4)*x_4^2 + (-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 4)*x_4*x_5 + 1/2*(-z_60^14 + z_60^6 + z_60^4)*x_5^2),
     x_0*x_1 + 1/3*x_2^2 +(1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 3)*x_0*x_3 + 1/4*(-6*z_60^14 + 6*z_60^6 + 6*z_60^4 + 3)*x_0*x_4 + 1/4*(-5*z_60^14 + 5*z_60^6 + 5*z_60^4 + 1)*x_0*x_5 + 1/4*(9*z_60^14 - 9*z_60^6- 9*z_60^4 - 6)*x_1*x_3 + 1/4*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 + 3)*x_1*x_4 + 1/4*(-7*z_60^14 + 7*z_60^6 + 7*z_60^4 + 3)*x_1*x_5 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 3)*x_2*x_3 + 1/4*(-8*z_60^14 + 8*z_60^6 + 8*z_60^4 + 3)*x_2*x_4 + 1/4*(-z_60^14 + z_60^6 +  z_60^4 + 1)*x_2*x_5) +c_5*(1/4*(z_60^14 - z_60^6 - z_60^4)*x_3^2 + (z_60^14 - z_60^6 - z_60^4 + 2)*x_3*x_4 + 1/2*(-6*z_60^14 + 6*z_60^6 + 6*z_60^4 - 9)*x_3*x_5 + 1/4*(-2*z_60^14 + 2*z_60^6 + 2*z_60^4 - 3)*x_4^2 + 1/2*(-2*z_60^14 + 2*z_60^6 + 2*z_60^4 - 3)*x_4*x_5 + 1/4*(-z_60^14 + z_60^6 + z_60^4 - 2)*x_5^2),
     x_0*x_2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 + 1)*x_2^2 +(1/4*(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0*x_3 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 3)*x_0*x_4 + 1/2*(-z_60^14 + z_60^6 + z_60^4)*x_0*x_5 + 1/4*(z_60^14 - z_60^6 - z_60^4 - 2)*x_1*x_4 + 1/4*(z_60^14 - z_60^6 - z_60^4 - 2)*x_1*x_5 + 1/4*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4)*x_2*x_3 +1/4*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_2*x_4 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 1)*x_2*x_5) +c_5*(1/2*(z_60^14 - z_60^6 - z_60^4 + 1)*x_3^2 + 1/2*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 + 5)*x_3*x_4 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 3)*x_3*x_5 + 1/4*(-z_60^14 + z_60^6 + z_60^4 - 1)*x_4^2 + 1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 - 7)*x_4*x_5 + 1/4*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 7)*x_5^2),
     x_1^2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 - 3)*x_2^2 +(1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 + 1)*x_0*x_3 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_0*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 3)*x_1*x_3 + 1/2*(z_60^14 - z_60^6 - z_60^4 - 1)*x_1*x_4 + 1/2*(z_60^14 - z_60^6 - z_60^4 - 1)*x_1*x_5 + 1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 + 1)*x_2*x_4 + 1/2*(-z_60^14 + z_60^6 + z_60^4 + 2)*x_2*x_5) +c_5*(1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 - 5)*x_3^2 + (2*z_60^14 - 2*z_60^6 - 2*z_60^4 + 2)*x_3*x_4 + (6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 9)*x_3*x_5 + (z_60^14 - z_60^6 - z_60^4 + 2)*x_4^2 + (2*z_60^14 - 2*z_60^6 - 2*z_60^4 + 4)*x_4*x_5 + 1/2*(5*z_60^14 - 5*z_60^6 - 5*z_60^4 + 8)*x_5^2),
     x_1*x_2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 - 2)*x_2^2 +(1/4*(-z_60^14 + z_60^6 + z_60^4 + 2)*x_0*x_3 + 1/4*(4*z_60^14 - 4*z_60^6 - 4*z_60^4 - 3)*x_0*x_5 + 1/4*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 3)*x_1*x_3 + 1/4*(z_60^14 - z_60^6 - z_60^4 + 1)*x_1*x_4 + 1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_1*x_5 + 1/4*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_2*x_4 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 1)*x_2*x_5) +c_5*(1/4*(-z_60^14 + z_60^6 + z_60^4 - 1)*x_3^2 + 1/2*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 + 5)*x_3*x_4 + 1/2*(6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 9)*x_3*x_5 + 1/2*(z_60^14 - z_60^6 - z_60^4 + 1)*x_4^2 + 1/2*(5*z_60^14 - 5*z_60^6 - 5*z_60^4 + 8)*x_4*x_5 + 1/4*(6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 11)*x_5^2)
     };

o6 : Ideal of S

i7 : time L=flatten entries gens gb(I,DegreeLimit=>4);
     -- used 0.278055 seconds

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_5-12*ww_60^14+12*ww_60^6+12*ww_60^4+15/2
c_2-(297/16)*ww_60^14+(297/16)*ww_60^6+(297/16)*ww_60^4+45/4
1

This suggests setting \( (c_5-12 \zeta_{60}^{14}+12\zeta_{60}^6+12\zeta_{60}^4+15/2)=0\).

This gives us a conjectural description of the desired pencil: we set \(c_5 = 12 \zeta_{60}^{14} - 12\zeta_{60}^6 - 12\zeta_{60}^4- 15/2)\) and let \(c_2 \) vary.

In the Magma session below, we check that least two values of \(c_2\) yield a nonsingular curve.

Checking the equations in `Magma`

We check that for two different values of \(c_{2}\), we obtain a smooth curve with the correct automorphisms. This implies that a general member of the pencil is a smooth curve with the correct automorphisms. However, I have not shown that the two curves studied below are not isomorphic to each other; it is possible that we have described a point in the moduli space \( \mathcal{M}_{6}\) rather than a curve in \( \mathcal{M}_6\). First, we study the value \(c_2 = 1\).

> K<z_60>:=CyclotomicField(60);
> P5<x_0,x_1,x_2,x_3,x_4,x_5>:=ProjectiveSpace(K,5);
> c_2:=1;
> c_5:=-(-12*z_60^14+12*z_60^6+12*z_60^4+15/2);
> X:=Scheme(P5,[x_0^2 + x_0*x_1 + (-z_60^14 + z_60^6 + z_60^4)*x_0*x_2 + x_1^2\
 + (-z_60^14 + z_60^6 + z_60^4 - 1)*x_1*x_2 + x_2^2+c_2*(x_3^2 + (z_60^14 - z_\
60^6 - z_60^4)*x_3*x_4 + (z_60^14 - z_60^6 - z_60^4)*x_3*x_5 + x_4^2 + x_4*x_5\
 + x_5^2),
> x_0^2 + 1/3*(z_60^14 - z_60^6 - z_60^4 - 2)*x_2^2 +(x_0*x_3 + 1/2*(-z_60^14 \
+ z_60^6 + z_60^4 + 1)*x_0*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_\
1*x_4 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_1*x_5 + 1/2*(3*z_60^14 - \
3*z_60^6 - 3*z_60^4 - 1)*x_2*x_4 + 1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 + 3)*\
x_2*x_5) +c_5*(x_3^2 + (-z_60^14 + z_60^6 + z_60^4 - 3)*x_3*x_4 + (3*z_60^14 -\
 3*z_60^6 - 3*z_60^4 + 3)*x_3*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 4)\
*x_4^2 + (-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 4)*x_4*x_5 + 1/2*(-z_60^14 + z_60\
^6 + z_60^4)*x_5^2),
> x_0*x_1 + 1/3*x_2^2 +(1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 3)*x_0*x_3 + 1/\
4*(-6*z_60^14 + 6*z_60^6 + 6*z_60^4 + 3)*x_0*x_4 + 1/4*(-5*z_60^14 + 5*z_60^6 \
+ 5*z_60^4 + 1)*x_0*x_5 + 1/4*(9*z_60^14 - 9*z_60^6- 9*z_60^4 - 6)*x_1*x_3 + 1\
/4*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 + 3)*x_1*x_4 + 1/4*(-7*z_60^14 + 7*z_60^6\
 + 7*z_60^4 + 3)*x_1*x_5 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 3)*x_2*x_3 +\
 1/4*(-8*z_60^14 + 8*z_60^6 + 8*z_60^4 + 3)*x_2*x_4 + 1/4*(-z_60^14 + z_60^6 +\
  z_60^4 + 1)*x_2*x_5) +c_5*(1/4*(z_60^14 - z_60^6 - z_60^4)*x_3^2 + (z_60^14 \
- z_60^6 - z_60^4 + 2)*x_3*x_4 + 1/2*(-6*z_60^14 + 6*z_60^6 + 6*z_60^4 - 9)*x_\
3*x_5 + 1/4*(-2*z_60^14 + 2*z_60^6 + 2*z_60^4 - 3)*x_4^2 + 1/2*(-2*z_60^14 + 2\
*z_60^6 + 2*z_60^4 - 3)*x_4*x_5 + 1/4*(-z_60^14 + z_60^6 + z_60^4 - 2)*x_5^2),
> x_0*x_2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 + 1)*x_2^2 +(1/4*(-z_60^14 + z_60^\
6 + z_60^4 - 1)*x_0*x_3 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 3)*x_0*x_4 + \
1/2*(-z_60^14 + z_60^6 + z_60^4)*x_0*x_5 + 1/4*(z_60^14 - z_60^6 - z_60^4 - 2)\
*x_1*x_4 + 1/4*(z_60^14 - z_60^6 - z_60^4 - 2)*x_1*x_5 + 1/4*(-3*z_60^14 + 3*z\
_60^6 + 3*z_60^4)*x_2*x_3 +1/4*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_2*x_4 +\
 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 1)*x_2*x_5) +c_5*(1/2*(z_60^14 - z_60^\
6 - z_60^4 + 1)*x_3^2 + 1/2*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 + 5)*x_3*x_4 + 1/\
2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 3)*x_3*x_5 + 1/4*(-z_60^14 + z_60^6 + z_\
60^4 - 1)*x_4^2 + 1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 - 7)*x_4*x_5 + 1/4*(-3\
*z_60^14 + 3*z_60^6 + 3*z_60^4 - 7)*x_5^2),
> x_1^2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 - 3)*x_2^2 +(1/2*(2*z_60^14 - 2*z_60\
^6 - 2*z_60^4 + 1)*x_0*x_3 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_0*x_\
5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 3)*x_1*x_3 + 1/2*(z_60^14 - z_60^6\
 - z_60^4 - 1)*x_1*x_4 + 1/2*(z_60^14 - z_60^6 - z_60^4 - 1)*x_1*x_5 + 1/2*(2*\
z_60^14 - 2*z_60^6 - 2*z_60^4 + 1)*x_2*x_4 + 1/2*(-z_60^14 + z_60^6 + z_60^4 +\
 2)*x_2*x_5) +c_5*(1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 - 5)*x_3^2 + (2*z_60^\
14 - 2*z_60^6 - 2*z_60^4 + 2)*x_3*x_4 + (6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 9)*\
x_3*x_5 + (z_60^14 - z_60^6 - z_60^4 + 2)*x_4^2 + (2*z_60^14 - 2*z_60^6 - 2*z_\
60^4 + 4)*x_4*x_5 + 1/2*(5*z_60^14 - 5*z_60^6 - 5*z_60^4 + 8)*x_5^2),
> x_1*x_2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 - 2)*x_2^2 +(1/4*(-z_60^14 + z_60^\
6 + z_60^4 + 2)*x_0*x_3 + 1/4*(4*z_60^14 - 4*z_60^6 - 4*z_60^4 - 3)*x_0*x_5 + \
1/4*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 3)*x_1*x_3 + 1/4*(z_60^14 - z_60^6 - z\
_60^4 + 1)*x_1*x_4 + 1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_1*x_5 + 1/4*(\
2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_2*x_4 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z\
_60^4 - 1)*x_2*x_5) +c_5*(1/4*(-z_60^14 + z_60^6 + z_60^4 - 1)*x_3^2 + 1/2*(3*\
z_60^14 - 3*z_60^6 - 3*z_60^4 + 5)*x_3*x_4 + 1/2*(6*z_60^14 - 6*z_60^6 - 6*z_6\
0^4 + 9)*x_3*x_5 + 1/2*(z_60^14 - z_60^6 - z_60^4 + 1)*x_4^2 + 1/2*(5*z_60^14 \
- 5*z_60^6 - 5*z_60^4 + 8)*x_4*x_5 + 1/4*(6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 11\
)*x_5^2)]);
> Dimension(X);
1
> IsSingular(X);
false
> HilbertPolynomial(Ideal(X));
10*$.1 - 5
2
> A:=Matrix([[z_60^14 - z_60^6 - z_60^4, 0, 1, 0, 0, 0],
> [0, -1, 0, 0, 0, 0],
> [z_60^14 - z_60^6 - z_60^4, 0, -z_60^14 + z_60^6 + z_60^4, 0, 0, 0],
> [0, 0, 0, -1, 0, 0],
> [0, 0, 0, 0, 0, -1],
> [0, 0, 0, 0, -1, 0]]);
> B:=Matrix([[0, -1, 0, 0, 0, 0],
> [-1, 0, 0, 0, 0, 0],
> [1, -1, -1, 0, 0, 0],
> [0, 0, 0, 0, 1, 0],
> [0, 0, 0, 1, 0, 0],
> [0, 0, 0, z_60^14 - z_60^6 - z_60^4, z_60^14 - z_60^6 - z_60^4, -1]]);
> C:=Matrix([[z_60^14-z_60^6-z_60^4, -z_60^14+z_60^6+z_60^4-1, -z_60^14+z_60^6\
+z_60^4-1, 0, 0, 0],
> [z_60^14-z_60^6-z_60^4,-z_60^14+z_60^6+z_60^4-1, -z_60^14+z_60^6+z_60^4, 0, \
0, 0],
> [-1, 1, 0, 0, 0, 0],
> [0, 0, 0, -1, 0, z_60^14-z_60^6-z_60^4],
> [0, 0, 0, 0, -1, 1],
> [0, 0, 0, 0, 0, 1]
> ]);
> Order(A);
2
> Order(B);
2
> Order(C);
2
> Order( (A*B*C)^(-1));
3
> GL6K:=GeneralLinearGroup(6,K);
> IdentifyGroup(sub<GL6K | A,B,C>);
<60, 5>
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
(z_60^14 - z_60^6 - z_60^4)*x_0 + (z_60^14 - z_60^6 - z_60^4)*x_2
-x_1
x_0 + (-z_60^14 + z_60^6 + z_60^4)*x_2
-x_3
-x_5
-x_4
and inverse
(z_60^14 - z_60^6 - z_60^4)*x_0 + (z_60^14 - z_60^6 - z_60^4)*x_2
-x_1
x_0 + (-z_60^14 + z_60^6 + z_60^4)*x_2
-x_3
-x_5
-x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_1 + x_2
-x_0 - x_2
-x_2
x_4 + (z_60^14 - z_60^6 - z_60^4)*x_5
x_3 + (z_60^14 - z_60^6 - z_60^4)*x_5
-x_5
and inverse
-x_1 + x_2
-x_0 - x_2
-x_2
x_4 + (z_60^14 - z_60^6 - z_60^4)*x_5
x_3 + (z_60^14 - z_60^6 - z_60^4)*x_5
-x_5
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
(z_60^14 - z_60^6 - z_60^4)*x_0 + (z_60^14 - z_60^6 - z_60^4)*x_1 - x_2
(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0 + (-z_60^14 + z_60^6 + z_60^4 - 1)*x_1 + 
    x_2
(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0 + (-z_60^14 + z_60^6 + z_60^4)*x_1
-x_3
-x_4
(z_60^14 - z_60^6 - z_60^4)*x_3 + x_4 + x_5
and inverse
(z_60^14 - z_60^6 - z_60^4)*x_0 + (z_60^14 - z_60^6 - z_60^4)*x_1 - x_2
(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0 + (-z_60^14 + z_60^6 + z_60^4 - 1)*x_1 + 
    x_2
(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0 + (-z_60^14 + z_60^6 + z_60^4)*x_1
-x_3
-x_4
(z_60^14 - z_60^6 - z_60^4)*x_3 + x_4 + x_5

Next, we study the value \(c_2 = 13 + \zeta_{60}^{29}\).

> K:=CyclotomicField(60);
> P5:=ProjectiveSpace(K,5);
> c_2:=13+z_60^29;
> c_5:=-(-12*z_60^14+12*z_60^6+12*z_60^4+15/2);
> X:=Scheme(P5,[x_0^2 + x_0*x_1 + (-z_60^14 + z_60^6 + z_60^4)*x_0*x_2 + x_1^2\
 + (-z_60^14 + z_60^6 + z_60^4 - 1)*x_1*x_2 + x_2^2+c_2*(x_3^2 + (z_60^14 - z_\
60^6 - z_60^4)*x_3*x_4 + (z_60^14 - z_60^6 - z_60^4)*x_3*x_5 + x_4^2 + x_4*x_5\
 + x_5^2),
> x_0^2 + 1/3*(z_60^14 - z_60^6 - z_60^4 - 2)*x_2^2 +(x_0*x_3 + 1/2*(-z_60^14 \
+ z_60^6 + z_60^4 + 1)*x_0*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_\
1*x_4 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_1*x_5 + 1/2*(3*z_60^14 - \
3*z_60^6 - 3*z_60^4 - 1)*x_2*x_4 + 1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 + 3)*\
x_2*x_5) +c_5*(x_3^2 + (-z_60^14 + z_60^6 + z_60^4 - 3)*x_3*x_4 + (3*z_60^14 -\
 3*z_60^6 - 3*z_60^4 + 3)*x_3*x_5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 4)\
*x_4^2 + (-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 4)*x_4*x_5 + 1/2*(-z_60^14 + z_60\
^6 + z_60^4)*x_5^2),
> x_0*x_1 + 1/3*x_2^2 +(1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 3)*x_0*x_3 + 1/\
4*(-6*z_60^14 + 6*z_60^6 + 6*z_60^4 + 3)*x_0*x_4 + 1/4*(-5*z_60^14 + 5*z_60^6 \
+ 5*z_60^4 + 1)*x_0*x_5 + 1/4*(9*z_60^14 - 9*z_60^6- 9*z_60^4 - 6)*x_1*x_3 + 1\
/4*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 + 3)*x_1*x_4 + 1/4*(-7*z_60^14 + 7*z_60^6\
 + 7*z_60^4 + 3)*x_1*x_5 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 3)*x_2*x_3 +\
 1/4*(-8*z_60^14 + 8*z_60^6 + 8*z_60^4 + 3)*x_2*x_4 + 1/4*(-z_60^14 + z_60^6 +\
  z_60^4 + 1)*x_2*x_5) +c_5*(1/4*(z_60^14 - z_60^6 - z_60^4)*x_3^2 + (z_60^14 \
- z_60^6 - z_60^4 + 2)*x_3*x_4 + 1/2*(-6*z_60^14 + 6*z_60^6 + 6*z_60^4 - 9)*x_\
3*x_5 + 1/4*(-2*z_60^14 + 2*z_60^6 + 2*z_60^4 - 3)*x_4^2 + 1/2*(-2*z_60^14 + 2\
*z_60^6 + 2*z_60^4 - 3)*x_4*x_5 + 1/4*(-z_60^14 + z_60^6 + z_60^4 - 2)*x_5^2),
> x_0*x_2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 + 1)*x_2^2 +(1/4*(-z_60^14 + z_60^\
6 + z_60^4 - 1)*x_0*x_3 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 3)*x_0*x_4 + \
1/2*(-z_60^14 + z_60^6 + z_60^4)*x_0*x_5 + 1/4*(z_60^14 - z_60^6 - z_60^4 - 2)\
*x_1*x_4 + 1/4*(z_60^14 - z_60^6 - z_60^4 - 2)*x_1*x_5 + 1/4*(-3*z_60^14 + 3*z\
_60^6 + 3*z_60^4)*x_2*x_3 +1/4*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_2*x_4 +\
 1/4*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 - 1)*x_2*x_5) +c_5*(1/2*(z_60^14 - z_60^\
6 - z_60^4 + 1)*x_3^2 + 1/2*(3*z_60^14 - 3*z_60^6 - 3*z_60^4 + 5)*x_3*x_4 + 1/\
2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 - 3)*x_3*x_5 + 1/4*(-z_60^14 + z_60^6 + z_\
60^4 - 1)*x_4^2 + 1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 - 7)*x_4*x_5 + 1/4*(-3\
*z_60^14 + 3*z_60^6 + 3*z_60^4 - 7)*x_5^2),
> x_1^2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 - 3)*x_2^2 +(1/2*(2*z_60^14 - 2*z_60\
^6 - 2*z_60^4 + 1)*x_0*x_3 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 1)*x_0*x_\
5 + 1/2*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 3)*x_1*x_3 + 1/2*(z_60^14 - z_60^6\
 - z_60^4 - 1)*x_1*x_4 + 1/2*(z_60^14 - z_60^6 - z_60^4 - 1)*x_1*x_5 + 1/2*(2*\
z_60^14 - 2*z_60^6 - 2*z_60^4 + 1)*x_2*x_4 + 1/2*(-z_60^14 + z_60^6 + z_60^4 +\
 2)*x_2*x_5) +c_5*(1/2*(-4*z_60^14 + 4*z_60^6 + 4*z_60^4 - 5)*x_3^2 + (2*z_60^\
14 - 2*z_60^6 - 2*z_60^4 + 2)*x_3*x_4 + (6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 9)*\
x_3*x_5 + (z_60^14 - z_60^6 - z_60^4 + 2)*x_4^2 + (2*z_60^14 - 2*z_60^6 - 2*z_\
60^4 + 4)*x_4*x_5 + 1/2*(5*z_60^14 - 5*z_60^6 - 5*z_60^4 + 8)*x_5^2),
> x_1*x_2 + 1/3*(-z_60^14 + z_60^6 + z_60^4 - 2)*x_2^2 +(1/4*(-z_60^14 + z_60^\
6 + z_60^4 + 2)*x_0*x_3 + 1/4*(4*z_60^14 - 4*z_60^6 - 4*z_60^4 - 3)*x_0*x_5 + \
1/4*(-3*z_60^14 + 3*z_60^6 + 3*z_60^4 + 3)*x_1*x_3 + 1/4*(z_60^14 - z_60^6 - z\
_60^4 + 1)*x_1*x_4 + 1/2*(2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_1*x_5 + 1/4*(\
2*z_60^14 - 2*z_60^6 - 2*z_60^4 - 1)*x_2*x_4 + 1/4*(3*z_60^14 - 3*z_60^6 - 3*z\
_60^4 - 1)*x_2*x_5) +c_5*(1/4*(-z_60^14 + z_60^6 + z_60^4 - 1)*x_3^2 + 1/2*(3*\
z_60^14 - 3*z_60^6 - 3*z_60^4 + 5)*x_3*x_4 + 1/2*(6*z_60^14 - 6*z_60^6 - 6*z_6\
0^4 + 9)*x_3*x_5 + 1/2*(z_60^14 - z_60^6 - z_60^4 + 1)*x_4^2 + 1/2*(5*z_60^14 \
- 5*z_60^6 - 5*z_60^4 + 8)*x_4*x_5 + 1/4*(6*z_60^14 - 6*z_60^6 - 6*z_60^4 + 11\
)*x_5^2)]);
> Dimension(X);
1
> IsSingular(X);
false
> Automorphism(X,A);
Mapping from: Sch: X to Sch: X
with equations : 
(z_60^14 - z_60^6 - z_60^4)*x_0 + (z_60^14 - z_60^6 - z_60^4)*x_2
-x_1
x_0 + (-z_60^14 + z_60^6 + z_60^4)*x_2
-x_3
-x_5
-x_4
and inverse
(z_60^14 - z_60^6 - z_60^4)*x_0 + (z_60^14 - z_60^6 - z_60^4)*x_2
-x_1
x_0 + (-z_60^14 + z_60^6 + z_60^4)*x_2
-x_3
-x_5
-x_4
> Automorphism(X,B);
Mapping from: Sch: X to Sch: X
with equations : 
-x_1 + x_2
-x_0 - x_2
-x_2
x_4 + (z_60^14 - z_60^6 - z_60^4)*x_5
x_3 + (z_60^14 - z_60^6 - z_60^4)*x_5
-x_5
and inverse
-x_1 + x_2
-x_0 - x_2
-x_2
x_4 + (z_60^14 - z_60^6 - z_60^4)*x_5
x_3 + (z_60^14 - z_60^6 - z_60^4)*x_5
-x_5
> Automorphism(X,C);
Mapping from: Sch: X to Sch: X
with equations : 
(z_60^14 - z_60^6 - z_60^4)*x_0 + (z_60^14 - z_60^6 - z_60^4)*x_1 - x_2
(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0 + (-z_60^14 + z_60^6 + z_60^4 - 1)*x_1 + 
    x_2
(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0 + (-z_60^14 + z_60^6 + z_60^4)*x_1
-x_3
-x_4
(z_60^14 - z_60^6 - z_60^4)*x_3 + x_4 + x_5
and inverse
(z_60^14 - z_60^6 - z_60^4)*x_0 + (z_60^14 - z_60^6 - z_60^4)*x_1 - x_2
(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0 + (-z_60^14 + z_60^6 + z_60^4 - 1)*x_1 + 
    x_2
(-z_60^14 + z_60^6 + z_60^4 - 1)*x_0 + (-z_60^14 + z_60^6 + z_60^4)*x_1
-x_3
-x_4
(z_60^14 - z_60^6 - z_60^4)*x_3 + x_4 + x_5

A pencil of genus 6 curves with automorphism group \(A_5\)

Obtaining candidate polynomials in Magma

Partial flattening stratification in Macaulay2

Checking the equations in Magma

Obtaining candidate polynomials in `Magma`

Partial flattening stratification in `Macaulay2`

Checking the equations in `Magma`