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 \(\langle P, v_p(d), f_p, c_p, K, s \rangle\) 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).