Tom Fisher's Home Page
Elements of order 5 in the TateShafarevich group

We list elements of order 5 in the TateShafarevich 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
http://www.warwick.ac.uk/~masgaj/ftp/data/
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
[a_{1},a_{2},
a_{3},a_{4},a_{6}]
of a minimal Weierstrass equation, the MordellWeil 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 (5^{k}  1)/2 genus one models of degree 5,
representing a subgroup of the 5Selmer 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 nonzero element of the subgroup
and its inverse.
We recall that a genus one model of degree 5 is a skewsymmetric
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
ϕ_{12}, ϕ_{13}, ϕ_{14}, ϕ_{15}, ϕ_{23}, ϕ_{24}, ϕ_{25}, ϕ_{34}, ϕ_{35}, ϕ_{45}.
We have checked that each subgroup of the Selmer group listed
has trivial intersection with the image of the MordellWeil group.
(In all examples so far the MordellWeil 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
F_{p}) 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
F_{p}points on the reduction mod p, and this
guarantees local solubility.
These tables were constructed using the following methods :

In most examples where E admits a rational 5isogeny, the
required elements of Sha are in the kernel of the map induced by
the 5isogeny. Explicit equations for each element of Sha
are then found by descent by 5isogeny, as described in [3] and [5].
In cases where E is a quadratic twist by d of one of
the curves C_{λ} defined in [3], we record the values
of λ and d.

In most examples where E does not admit a rational 5isogeny,
we find that the 5torsion 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
with MordellWeil rank 2 or 3. We specify F in the second
row of each entry, giving the data in the same format as for E.
In some cases F is beyond the range of Cremona's tables.
Explicit equations for each element of Sha are then computed using the
Hessian [4], and related covariants [6],
as contributed to MAGMA [8] by the author.
The same formulae allows us to compute elements of Sha for
E from those for E' where E and E'
are related by an isogeny of degree coprime to 5.

In the remaining examples we used full 5descent,
as described in [2] and [5], and contributed to MAGMA [8] by the author.
If possible we first used visibility
(usually with F a curve of rank 1) to find some elements of
Sha, and then used 5descent to "add" these examples. This left a
comparatively small number of examples where we needed to
compute the Sunits in a degree 24 number field. The latter were
computed by Steve Donnelly in September 2014.
A similar study of elements of Sha of order 3 is
here.
References

[1] John Cremona, Barry Mazur,
Visualizing elements in the ShafarevichTate group,
Experiment. Math. 9 (2000), no. 1, 1328.

[2] John Cremona, Tom Fisher, Cathy O'Neil, Denis Simon, Michael Stoll,
Explicit ndescent on elliptic curves,
I. Algebra, J. reine angew. Math. 615 (2008) 121155.
II. Geometry, J. reine angew. Math. 632 (2009) 6384.
III. Algorithms.

[3] T.A. Fisher, On 5 and 7 descents for elliptic curves, PhD thesis, University of Cambridge, 2000.

[4] T.A. Fisher, The Hessian of a genus one curve, Proc. Lond. Math. Soc. (3) 104 (2012) 613648.

[5] T.A. Fisher, Explicit 5descent on elliptic curves, in ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, E.W. Howe, K.S. Kedlaya (eds.),
Open Book Series, 1, Mathematical Sciences Publishers, Berkeley, 2013.

[6] T.A. Fisher Invariant theory for the elliptic normal quintic, I. Twists of X(5),
Math. Ann. 356 (2013), no. 2, 589616.

[7] T.A. Fisher, Minimisation and reduction of 5coverings of elliptic curves,
Algebra & Number Theory 7 (2013), no. 5, 1179–1205.

[8] MAGMA is described in
W. Bosma, J. Cannon, C. Playoust,
The Magma algebra system I: The user language,
J. Symbolic Comput. 24, 235265 (1997).
The MAGMA home page is at
http://magma.maths.usyd.edu.au/magma/.
Last updated 16th February 2015