Journal of the Ramanujan Mathematical Society

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On Genus One Curves of Degree 5 With Square-Free Discriminant


Affiliations
1 University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
2 American University in Cairo, Mathematics and Actuarial Science Department, AUC Avenue, New Cairo, Egypt
     

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We study genus one curves of degree 5 defined by Pfaffians. We give new formulae for the invariants, and prove the equivalence of two different definitions of minimality. As an application we show that transformations between models with square-free discriminant are necessarily integral. This result is used by Bhargava and Shankar in their work on the average ranks of elliptic curves.
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  • On Genus One Curves of Degree 5 With Square-Free Discriminant

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Authors

Tom Fisher
University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
Mohammad Sadek
American University in Cairo, Mathematics and Actuarial Science Department, AUC Avenue, New Cairo, Egypt

Abstract


We study genus one curves of degree 5 defined by Pfaffians. We give new formulae for the invariants, and prove the equivalence of two different definitions of minimality. As an application we show that transformations between models with square-free discriminant are necessarily integral. This result is used by Bhargava and Shankar in their work on the average ranks of elliptic curves.