Optimal curves differing by a 5-isogeny

Dongho Byeon, and Taekyung Kim. "Optimal curves differing by a 5-isogeny." Acta Arithmetica 165.4 (2014): 351-359. <http://eudml.org/doc/279350>.

@article{DonghoByeon2014,
abstract = {For i = 0,1, let $E_i$ be the $X_i(N)$-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.},
author = {Dongho Byeon, Taekyung Kim},
journal = {Acta Arithmetica},
keywords = {elliptic curves; optimal curves; isogeny},
language = {eng},
number = {4},
pages = {351-359},
title = {Optimal curves differing by a 5-isogeny},
url = {http://eudml.org/doc/279350},
volume = {165},
year = {2014},
}

TY - JOUR
AU - Dongho Byeon
AU - Taekyung Kim
TI - Optimal curves differing by a 5-isogeny
JO - Acta Arithmetica
PY - 2014
VL - 165
IS - 4
SP - 351
EP - 359
AB - For i = 0,1, let $E_i$ be the $X_i(N)$-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.
LA - eng
KW - elliptic curves; optimal curves; isogeny
UR - http://eudml.org/doc/279350
ER -