Curves over p-adic Fields

The functions in this section are for elliptic curves defined over p-adic fields. They provide an interface to the same code for Tate's algorithm that is used for curves over number fields.

Local Invariants

Conductor(E) : CrvEll -> FldPadElt

The conductor of the elliptic curve E defined over a p-adic field.

LocalInformation(E) : CrvEll -> Tup, CrvEll

Implements Tate's algorithm for the elliptic curve E over a p-adic field. This intrinsic computes local reduction data and a local minimal model. The model is not required to be integral on input. Output is < P, v_p(d), f_p, c_p, K, s > and E_min where P is the uniformizer of the ground field, v_p(d) is the valuation of the local minimal discriminant, f_p is the valuation of the conductor, c_p is the Tamagawa number, K is the Kodaira Symbol, and s is false if the curve has non-split multiplicative reduction and true otherwise. E_min is an integral minimal model of E.

RootNumber(E) : CrvEll -> RngIntElt

The local root number of the elliptic curve E (defined over a p-adic field).