# On The Rank Of Congruent Elliptic Curves

## TL;DR

This paper constructs specific congruent elliptic curves related to primes and determines their ranks in cases where they are congruent numbers, advancing understanding of their algebraic properties.

## Contribution

It introduces a method to construct congruent elliptic curves for various prime-based cases and determines their ranks when they are congruent numbers.

## Key findings

- Ranks are explicitly determined for constructed curves in congruent number cases.
- The paper provides new insights into the algebraic structure of these elliptic curves.
- Results contribute to the classification of ranks for prime-related congruent elliptic curves.

## Abstract

In this paper, $p$ and $q$ are two different odd primes. First, We construct the congruent elliptic curves corresponding to $p$, $2p$, $pq$, and $2pq,$ then, in the cases of congruent numbers, we determine the rank of the corresponding congruent elliptic curves.