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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 10000 (to be continued ...)

• 1-9999

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 12 genus one models of degree 5, representing a subgroup of the 5-Selmer group of E isomorphic to Z/5Z x Z/5Z. 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 &phi of linear forms in 5 variables. The corresponding curve is defined by the 4 × 4 Pfaffians of &phi. We convert &phi to a sequence of 50 coefficients by taking the linear forms in the order

In all examples so far, we have E(Q)/5E(Q) = 0. Therefore each curve listed is a counterexample to the Hasse Principle.

These tables were constructed using the following methods :-

• In all examples where E admits a rational 5-isogeny, we find that E is a quadratic twist (by d) of one of the curves C_lambda defined in [3]. Explicit equations for each element of Sha are then found by descent by 5-isogeny, as described in [3].
• In most examples where E does not admit a rational 5-isogeny, we find that the 5-torsion of Sha is visible (in the sense of Mazur [1]) in an abelian surface isogenous to E × F where F is an elliptic curve over Q of Mordell-Weil rank 2. We specify F in the second row of each entry, again taken directly from Cremona's tables. Explicit equations for each element of Sha are then computed using the Hessian (see [4]) and related covariants, as contributed to MAGMA [5] by the author.
• In the remaining examples (of conductor 6471, 6516 and 6727) we used full 5-descent, as described in [2]. In each case we used PARI [6] to find a set of fundamental units in a degree 24 number field. The remainder of the calculation was then performed using programs written by Michael Stoll and myself in MAGMA [5].

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

References

• [1] John Cremona, Barry Mazur, Visualizing elements in the Shafarevich-Tate group, Experiment. Math. 9 (2000), no. 1, 13--28.
• [2] John Cremona, Tom Fisher, Cathy O'Neil, Denis Simon, Michael Stoll, Explicit n-descent on elliptic curves, I. Algebra, II. Geometry, III Algorithms.
• [3] T.A. Fisher, On 5 and 7 descents for elliptic curves, PhD thesis.
• [4] T.A. Fisher, The Hessian of a genus one curve, preprint.
• [5] MAGMA is described in W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24, 235--265 (1997). The MAGMA home page is at http://magma.maths.usyd.edu.au/magma/.
• [6] PARI/GP is developped by the PARI Group, University of Bordeaux. The PARI home page is at http://pari.math.u-bordeaux.fr/.