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

Watkins’s 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 a rational point of order 2 and prime power conductor, in particular, for the congruent number elliptic curves. Furthermore, we give a lower bound for the congruence number for elliptic curves of the form y 2 =x 3 -dx, with d a fourth power free integer.