Optimal curves differing by a 5-isogeny
Tom 165 / 2014
Acta Arithmetica 165 (2014), 351-359
MSC: Primary 11G05; Secondary 14K02.
DOI: 10.4064/aa165-4-5
Streszczenie
For , let E_i be the X_i(N)-optimal curve of an isogeny class \mathcal {C} of elliptic curves defined over \mathbb Q of conductor N. Stein and Watkins conjectured that E_0 and E_1 differ by a 5-isogeny if and only if E_0=X_0(11) and E_1=X_1(11). In this paper, we show that this conjecture is true if N is square-free and is not divisible by 5. On the other hand, Hadano conjectured that for an elliptic curve E defined over \mathbb Q with a rational point P of order 5, the 5-isogenous curve E':=E/\langle P \rangle has a rational point of order 5 again if and only if E'=X_0(11) and E=X_1(11). In the process of the proof of Stein and Watkins's conjecture, we show that Hadano's conjecture is not true.
