Fwd: Remarks on isogenies (Re: Table of Elliptic Curves over Q(sqrt(5))
William Stein
From: Noam Elkies <elk...@math.harvard.edu>
Date: Tue, Aug 16, 2011 at 4:50 PM
Subject: Remarks on isogenies (Re: Table of Elliptic Curves over Q(sqrt(5))
To: wst...@gmail.com
Cc: T.A.F...@dpmms.cam.ac.uk, elle...@math.wisc.edu,
john.c...@gmail.com, jvo...@gmail.com,
uwntr...@googlegroups.com
P.S. I wrote:
> [...]
> The "remarks on isogenies" preceding the Antwerp tables (and apparently
> not included in the scans at modular.math.washington.edu/Tables/antwerp :-( )
> list the other 10 curves X_0(N) of genus 1; most of them still have
> rank zero over Q(sqrt(5)), but there are two more cases with rank 1,
> [...]
When 5|N it is also possible for the rank to remain 1 but with a larger
torsion group, as happened for X_1(15). The same is true for X_0(15)
(Antwerp 15C, and 45a:A4 in your new table), which has 8 rational points
but 16 over Q(sqrt(5)). The rational points are described in the Antwerp
"remarks on isogenies": there are two orbits under <w_3,w_5>, one consisting
of the four cusps, the other giving rise to the 15-isogeny at level 50 --
two isogeny classes over Q, but related by sqrt(5) twist and thus
isomorphic over Q(sqrt(5)), and including the curve with 15-torsion.
The new points all give rise to 15-isogenies between CM curves that
seem to be just beyond the range of your table. There are two orbits
of size 2, coming from fixed points of w_15; these give rise to elliptic
curves with CM discriminant -15 and -60, and will form an isogeny class
featuring 30-isogenies, a phenomenon that cannot happen over Q.
The remaining four points form a single orbit and give rise to isogenies
between CM curves of discriminant -3 and -75. (The CM-3 curve must be
isomorphic over Q(sqrt(5)) with [0,0,1,0,1] to have 5-isogenies defined
over that field.) Thus this class features 75-isogenies, again not seen
over Q. I think both of these must have conductor norm at least
3^4 5^2 = 2025, so accessible to your techniques but just beyond
the range of this summer's computation (unless William's OK with
going out to 25 more conductor norms to reach a bunch of exotic
CM curves and isogenies).
Something like this happens again for N=20, but with a different outcome.
Here X_0(20) is [0,1,0,4,4], Antwerp 20B and 80a:A4 in your tables, and
with M-W group Z/6Z over either Q or Q(sqrt(5)). But the 2-isogenous curve
[0,1,0,-1,0] (20A / 80a:A1), which parametrizes elliptic curves with a
subgroup Z/2Z x Z/10Z, has Z/6Z over Q but Z/2Z x Z/6Z over Q(sqrt(5)).
Nevertheless there are no such curves over Q(sqrt(5)), because all 12
of these points are cusps. Gamma_1(5) has two rational cusps and
two conjugate over Q(sqrt(5)); taking the intersection with Gamma(2)
(which has three cusps, all rational) yields six cusps of each kind,
with the rational cusps giving the rational points over Q and the
irrational cusps giving the new rational points over Q(sqrt(5)).
NDE
--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org