Analytic order of Ш (reviewed)

The Tate-Shafarevic group Ш of an elliptic curve $E$ defined over $\mathbb{Q}$ is a torsion group defined in terms of Galois cohomology, which is conjectured to be finite. Its order #Ш appears in the strong form of the Birch-Swinnerton-Dyer Conjecture for $E$. The value of the order which is predicted by the conjecture is called the Analytic Order of Sha, Ш_an. Note that Ш_an predicted by the conjecture is always a square.

For elliptic curves of rank $0$ or $1$ it is a theorem that Ш_an is a rational number, and this rational number has been computed exactly for each curve in the database. These values are always in fact integer squares in all cases computed to date. For curves of rank $2$ and above, there is no such theorem, and the values computed are simply floating point approximate values which happen to be very close to integers. In the LMFDB we store the rounded values in this case.