# On $p$-degree of elliptic curves

## TL;DR

This paper studies the $p$-degree function of elliptic curves over $Q_p$, analyzing its properties and computing it explicitly in certain cases to understand the complexity of $p$-torsion points.

## Contribution

It introduces and investigates the $p$-degree function for elliptic curves over $Q_p$, providing new properties and explicit computations in special cases.

## Key findings

- Properties of the $p$-degree function established
- Explicit calculations for specific elliptic curves
- Insights into the complexity of $p$-torsion points

## Abstract

In this note we investigate the $p$-degree function of elliptic curves over the field $\mathbb{Q}_p$ of $p$-adic numbers. The $p$-degree measures the least complexity of a non-zero $p$-torsion point on an elliptic curve. We prove some properties of this function and compute it explicitly in some special cases.