Why does MAGMA claim that the automorphism group of a curve is trivial?

Question

I have been trying to compute the Automorphism group of a curve using MAGMA with no success. This is what I have tried: I have tried to compute the Automorphism group of the curve $y^3=x^4-x$ and no matter what I try -- it produces a trivial AutomorphismsGroup over rationals and I am not able to extend the scalars.

Attempt 1:


K:=Rationals(); 
P<x,y,z>:=ProjectiveSpace(K,2);
C:=Curve(P,y^3*z-x^4+x*z^3);
G:=AutomorphismGroup(C);
#G;
1

The group is trivial... However, we know that over the complex numbers the group is $C_9$, cyclic group of order 9. But, I am not able to extend the scalars to complex numbers and get this result...

L:=AlgebraicClosure(K);
D:=BaseChange(C,L);
G:=AutomorphismGroup(D);
                       ^
Runtime error in 'AutomorphismGroup': Curve must have a function field

or

L:=ComplexField();
D:=BaseChange(C,L); 
                ^
Runtime error in 'BaseChange': Ring must be exact

I have tried to compute the group via function fields:

Attempt 2:
A<x>:=PolynomialRing(K);
B<y>:=PolynomialRing(A);
F:=FunctionField(y^3-x^4+x);
G:=AutomorphismGroup(F);
#G;
1

and 
L:=ComplexField();
BaseChange(F,L);
             ^
Runtime error in 'BaseChange': Bad argument types
Argument types given: FldFun[FldFunRat[FldRat]], FldCom

First question: How can one compute the Automorphism group of the curve over the complex numbers using MAGMA?

Second question: Is it possible to compute (in MAGMA) the Automorphism group of the Jacobian and then, using the Torelli theorem, the automorphism group of a curve?

Answer 1

AutomorphismGroup computes the automorphism group over the base field, and it only works with certain types of base fields - in particular, it won't work over the reals, complexes, and the "algebraic closure" (none of which are really suited to geometric computations). But you can get what you want by working over a number field, and in your example it's not hard to see which one you need:

> K := CyclotomicField(3);
> P<x,y,z> := ProjectiveSpace(K, 2);
> C := Curve(P, y^3*z - x^4 + x*z^3);
> G := AutomorphismGroup(C);
> #G;
3

This doesn't really help if you can't work out the field of definition of the automorphisms you want in advance. If you're working with hyperelliptic curves, though, you can use GeometricAutomorphismGroup to get the automorphism group over the algebraic closure.