Parameterized Families of Elliptic Curves with Large Rational Torsion Subgroups

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The set of rational points on an elliptic curve
defined over the rationals
with at least one rational point
is endowed with a group law that can be described geometrically using the chord-and-tangent method. Further, it is a well-known result that if
is a rational point of order
for
, then
is birationally equivalent to an elliptic curve with an equation
, where
and
is a rational point of order
. That is, all elliptic curves
with a rational point of order
are in a one-parameter family if
.
Contributed by: Christopher Grattoni (June 2015)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Snapshot 1: this elliptic curve has a rational point of order 8, and a geometric representation of the group law is visible
Snapshot 2: this elliptic curve has a rational point of order 5
Snapshot 3: this elliptic curve has a rational point of order 12, and a geometric representation of the group law is visible
The notation for an elliptic curve
is the multiplication-by-
map on
. That is, if we call the group law on the elliptic curve "addition," then
is defined as "adding"
to itself
times. The identity element for this group law is the point at infinity
for the projective plane. Two points sum to
when they intersect the same vertical line. The yellow lines in the Demonstration are the lines that arise from repeatedly adding
to itself. The vertical orange line is meant to signify the final sum,
. The Demonstration "Addition of Points on an Elliptic Curve over the Reals" shows this chord-and-tangent formulation of the elliptic curve group law.
You may want to know how to compute such that
has a rational point of order
at the point
. First, suppose
has a rational point of order
. This implies that
is birationally equivalent to an elliptic curve in Tate normal form,
for
, such that
is a rational point of order
. Next, compute
and suppose it equals
. This allows one to find a relation between
and
. Once this relation is found, use this in
, which results in the desired one-parameter family of curves with a rational point of order
for
. These results are summarized below:
References
[1] J. H. Silverman, The Arithmetic of Elliptic Curves, New York: Springer-Verlag, 1986.
[2] I. García, M. A. Olalla, and J. M. Tornero, "Computing the Rational Torsion of an Elliptic Curve Using Tate Normal Form," Journal of Number Theory, 96(1), 2002 pp. 76–88. doi:10.1006/jnth.2002.2780.
[3] E. V. Flynn and C. Grattoni, "Descent via Isogeny on Elliptic Curves with Large Rational Torsion Subgroups," Journal of Symbolic Computation, 43(4), 2008 pp. 293–303. doi:10.1016/j.jsc.2007.11.001.
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