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Are there are any references that record the cardinality of Ш for elliptic curves for which Ш is known? Also their corresponding conductors.

EDIT: Following the Qiaochu Yuan's comment's I should mention that Ш is the Tate-Shafarevich group.

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Ш doesn't make a great search term, so for the sake of completeness I'll mention that Eugene is asking about the Tate-Shafarevich group: en.wikipedia.org/wiki/Tate%E2%80%93Shafarevich_group –  Qiaochu Yuan May 26 '12 at 15:22

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The Cremona database, in particular table four, lists the analytic order of sha for optimal curves, i.e. the order as predicted by the Birch and Swinnerton-Dyer conjecture. For curves of analytic rank $\leq 1$, this is known to always be the correct value up to a few primes. But thanks to calculations of Stein and Wuthrich, this is known to be the exact correct values for all curves of rank $\leq 1$ and conductor up to 130000 (might be more by now).

For curves of rank $\geq 2$, our state of knowledge is extremely hazy. In particular, there is not a single elliptic curve over $\mathbb{Q}$, for which sha is proven to be finite! This is in stark contrast to what we expect to be true: a conjecture of Delauney, a version of the Cohen-Lenstra heuristics for elliptic curves, predicts that a huge majority of curves of rank $\geq 2$ (I believe, more than 98%) should have trivial Tate-Shafarevich group. Cremona's table lists four curves of conductor up to 230000 with rank 2 and with non-trivial sha, and the remaining ones of rank 2 up to that conductor should have trivial sha (but as I say, we don't even know if their Tate-Shafarevich groups are finite).

In this paper by Stein and Wuthrich, the $p$-primary part of sha is computed (and shown to be trivial) for lots of curves and primes $p$, in particular for lots of curves of higher rank. Their computations are unconditional, i.e. don't depend on the BSD.

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I had a feeling that Cremona would be the answer but my google searches came up blank (I'm probably not that good at search terms). Thanks! –  Eugene May 27 '12 at 4:18
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You are welcome! I recommend bookmarking that page that contains Cremona's tables. It's an extremely useful resource! –  Alex B. May 27 '12 at 4:21
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After taking a close look at it I suspect I have died and gone to heaven! –  Eugene May 27 '12 at 4:25
    
:-) With your permission, I will quote you when I next talk to Cremona. –  Alex B. May 27 '12 at 4:27
    
Not sure if it's worth much coming from someone like me but feel free! Thanks again for the link! –  Eugene May 27 '12 at 4:32

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