Jacobians and ranks of a curve

Question

I would like to know the following:

How to find Jacobian and rank of an hyper elliptic curve like $x^5-x= y^2-y$?

High regards

Rosy

Answer 1

For what it's worth, I used Magma to compute the rank and torsion subgroup of the jacobian, as follows:

P<x>:=PolynomialRing(Rationals());
C:=HyperellipticCurve(x^5-x,-1);
J:=Jacobian(C);
RankBounds(J);
C:=HyperellipticCurve(4*(x^5-x-1/4),0);
TorsionSubgroup(Jacobian(C));

For the torsion, Magma requires the curve to be of the form $y^2=f(x)$, so I changed variables and wrote the curve as $y^2=4x^5-4x-1$. The answer is $J\cong \mathbb{Z}^3$.

If you want to do this by hand, then I recommend the linked article by Michael Stoll.