# 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).