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Elements of order 5 in the Tate-Shafarevich group

We list elements of order 5 in the Tate-Shafarevich group for elliptic curves E over Q. To compile this data we started from Cremona's list of elliptic curves over Q with "analytic order of Sha" greater than 1. The data is available at

For each elliptic curve over Q with analytic order of Sha divisible by 5 we expect to find a subgroup of Sha isomorphic to Z/5Z x Z/5Z. We have found such a subgroup in all cases up to conductor 350000 (the current limit of Cremona's tables).

The first line of each entry specifies an elliptic curve E over Q. This data is taken directly from Cremona's tables. The quantities listed are the conductor, the isogeny class, the number in the isogeny class, the coefficients [a1,a2, a3,a4,a6] of a minimal Weierstrass equation, the Mordell-Weil rank, the order of the torsion subgroup, and finally the analytic order of Sha. The next line gives some data relating to how the example was computed. We then list (5k - 1)/2 genus one models of degree 5, representing a subgroup of the 5-Selmer group of E isomorphic to (Z/5Z)k where k = 2 or 4 (the latter so far only in the case 165066d3). Each genus one model corresponds to a non-zero element of the subgroup and its inverse.

We recall that a genus one model of degree 5 is a skew-symmetric matrix ϕ of linear forms in 5 variables. The corresponding curve is defined by the 4 × 4 Pfaffians of ϕ. We convert ϕ to a sequence of 50 coefficients by taking the linear forms in the order

ϕ12, ϕ13, ϕ14, ϕ15, ϕ23, ϕ24, ϕ25, ϕ34, ϕ35, ϕ45.

We have checked that each subgroup of the Selmer group listed has trivial intersection with the image of the Mordell-Weil group. (In all examples so far the Mordell-Weil rank is 0 or 1.) Therefore each curve listed is a counterexample to the Hasse Principle.

The genus one models listed have been minimised and reduced as described in [7]. We have also arranged that the reduction mod p at each prime p is geometrically (i.e. over the algebraic closure of Fp) either an irreducible curve of degree 5, or (in some cases where the Tamagawa number of E at p is divisible by 5) the union of a rational normal curve of degree 4 and a line. In particular there are always smooth Fp-points on the reduction mod p, and this guarantees local solubility.

These tables were constructed using the following methods :-

A similar study of elements of Sha of order 3 is here.


Last updated 16th February 2015