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• # Artykuł - szczegóły

## Acta Arithmetica

2014 | 165 | 4 | 351-359

## Optimal curves differing by a 5-isogeny

EN

### Abstrakty

EN
For i = 0,1, let $E_i$ be the $X_i(N)$-optimal curve of an isogeny class 𝓒 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.

351-359

wydano
2014

### Twórcy

autor
• Department of Mathematics, Seoul National University, Seoul, Korea
autor
• Department of Mathematics, Seoul National University, Seoul, Korea