For i = 0,1, let be the -optimal curve of an isogeny class of elliptic curves defined over ℚ of conductor N. Stein and Watkins conjectured that E₀ and E₁ differ by a 5-isogeny if and only if E₀ = X₀(11) and E₁ = X₁(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 ℚ with a rational point P of order 5, the 5-isogenous curve E’ := E/⟨P⟩ has a rational point of order 5 again if and only if E’ = X₀(11) and E = X₁(11). In the process of the proof of Stein and Watkins’s conjecture, we show that Hadano’s conjecture is not true.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-4-5,
author = {Dongho Byeon and Taekyung Kim},
title = {Optimal curves differing by a 5-isogeny},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {351-359},
zbl = {1331.11041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-4-5}
}
Dongho Byeon; Taekyung Kim. Optimal curves differing by a 5-isogeny. Acta Arithmetica, Tome 166 (2014) pp. 351-359. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-4-5/