Watkins's conjecture for quadratic twists of Elliptic Curves with Prime Power Conductor

Abstract

Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the 2-adic valuation of its modular degree. We show that this conjecture is satisfied when E is any quadratic twist of an elliptic curve with rational 2-torsion and prime power conductor. Furthermore, we give a lower bound of the congruence number for elliptic curves of the form y 2 = x 3 − dx, with d a biquadratefree integer.