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Optimal curves differing by a 5-isogeny

Dongho Byeon, Taekyung Kim (2014)

Acta Arithmetica

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...

The analytic order of III for modular elliptic curves

J. E. Cremona (1993)

Journal de théorie des nombres de Bordeaux

In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny class.

Volcanoes of l-isogenies of elliptic curves over finite fields: The case l=3.

Josep M. Miret Biosca, Daniel Sadornil Renedo, Juan Tena Ayuso, Rosana Tomàs, Magda Valls Marsal (2007)

Publicacions Matemàtiques

This paper is devoted to the study of the volcanoes of ℓ-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the ℓ-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case ℓ = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results...

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