Tom Fisher's Home Page
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
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 .
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 :-
In most examples where E admits a rational 5-isogeny, the
required elements of Sha are in the kernel of the map induced by
the 5-isogeny. Explicit equations for each element of Sha
are then found by descent by 5-isogeny, as described in  and .
In cases where E is a quadratic twist by d of one of
the curves Cλ defined in , we record the values
of λ and d.
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 )
in an abelian surface isogenous to E × F,
where F is an elliptic curve over Q
with Mordell-Weil 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 , and related covariants ,
as contributed to MAGMA  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 5-descent,
as described in  and , and contributed to MAGMA  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 5-descent to "add" these examples. This left a
comparatively small number of examples where we needed to
compute the S-units 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
 John Cremona, Barry Mazur,
Visualizing elements in the Shafarevich-Tate group,
Experiment. Math. 9 (2000), no. 1, 13--28.
 John Cremona, Tom Fisher, Cathy O'Neil, Denis Simon, Michael Stoll,
Explicit n-descent on elliptic curves,
I. Algebra, J. reine angew. Math. 615 (2008) 121-155.
II. Geometry, J. reine angew. Math. 632 (2009) 63-84.
 T.A. Fisher, On 5 and 7 descents for elliptic curves, PhD thesis, University of Cambridge, 2000.
 T.A. Fisher, The Hessian of a genus one curve, Proc. Lond. Math. Soc. (3) 104 (2012) 613-648.
 T.A. Fisher, Explicit 5-descent 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.
 T.A. Fisher Invariant theory for the elliptic normal quintic, I. Twists of X(5),
Math. Ann. 356 (2013), no. 2, 589-616.
 T.A. Fisher, Minimisation and reduction of 5-coverings of elliptic curves,
Algebra & Number Theory 7 (2013), no. 5, 1179–1205.
 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
Last updated 16th February 2015