# Elliptic curves¶

Maps between them

## Elliptic curves over number fields¶

- Elliptic curves over the rational numbers
- Tables of elliptic curves of given rank
- Elliptic curves over number fields
- Canonical heights for elliptic curves over number fields
- Saturation of Mordell-Weil groups of elliptic curves over number fields
- Torsion subgroups of elliptic curves over number fields (including \(\QQ\))
- Galois representations attached to elliptic curves
- Galois representations for elliptic curves over number fields
- Isogeny class of elliptic curves over number fields
- Tate-Shafarevich group
- Complex multiplication for elliptic curves
- Testing whether elliptic curves over number fields are \(\QQ\)-curves

The following relate to elliptic curves over local nonarchimedean fields.

Analytic properties over \(\CC\).

Modularity and \(L\)-series over \(\QQ\).

# Hyperelliptic curves¶

- Hyperelliptic curve constructor
- Hyperelliptic curves over a general ring
- Hyperelliptic curves over a finite field
- Hyperelliptic curves over a \(p\)-adic field
- Hyperelliptic curves over the rationals
- Mestre’s algorithm
- Computation of Frobenius matrix on Monsky-Washnitzer cohomology
- Frobenius on Monsky-Washnitzer cohomology of a hyperelliptic curve over a
- Jacobian of a general hyperelliptic curve
- Jacobian of a hyperelliptic curve of genus 2
- Rational point sets on a Jacobian
- Jacobian ‘morphism’ as a class in the Picard group
- Hyperelliptic curves of genus 2 over a general ring
- Compute invariants of quintics and sextics via ‘Ueberschiebung’
- Kummer surfaces over a general ring
- Conductor and reduction types for genus 2 curves