The analytic order of III for modular elliptic curves
Journal de théorie des nombres de Bordeaux, Volume 5 (1993) no. 1, pp. 179-184.

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.

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author = {J. E. Cremona},
title = {The analytic order of {III} for modular elliptic curves},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {179--184},
publisher = {Universit\'e Bordeaux I},
volume = {5},
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year = {1993},
zbl = {0795.14016},
mrnumber = {1251236},
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J. E. Cremona. The analytic order of III for modular elliptic curves. Journal de théorie des nombres de Bordeaux, Volume 5 (1993) no. 1, pp. 179-184. https://jtnb.centre-mersenne.org/item/JTNB_1993__5_1_179_0/

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[2] A. Brumer and O. Mcguinness, The behaviour of the Mordell-Weil group of elliptic curves, Bull. AMS (New Series) 23 (1990), 375-382. | MR | Zbl

[3] J.W.S. Cassels, Arithmetic on curves of genus 1 (VIII). On the conjectures of Birch and Swinnerton-Dyer, J. Reine Angew. Math. 217 (1965), 180-189. | MR | Zbl

[4] J.E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press 1992. | MR | Zbl

[5] V.I. Kolyvagin, Finiteness of E(Q) and IIIE/Q for a subclass of Weil curves, Math. USSR Izvest. 32 (1989), 523-542. | MR | Zbl

[6] J. Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris, sér. A 273 (1971), 238-241. | MR | Zbl