# On a special case of Watkins' conjecture

@article{Kazalicki2016OnAS, title={On a special case of Watkins' conjecture}, author={Matija Kazalicki and Daniela Kohen}, journal={arXiv: Number Theory}, year={2016} }

Watkins' conjecture asserts that for a rational elliptic curve $E$ the degree of the modular parametrization is divisible by $2^r$, where $r$ is the rank of $E$. In this paper we prove that if the modular degree is odd then $E$ has rank $0$. Moreover, we prove that the conjecture holds for all rank two rational elliptic curves of prime conductor and positive discriminant.

## 3 Citations

### A conjecture of Watkins for quadratic twists

- Mathematics
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Watkins conjectured that for an elliptic curve E over Q of Mordell-Weil rank r, the modular degree of E is divisible by 2. If E has non-trivial rational 2-torsion, we prove the conjecture for all the…

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

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. 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…

### Corrigendum to “On a special case of Watkins’ conjecture”

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