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 .

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In this Demonstration, you can pick from a torsion subgroup of order and select integer values for the parameter to vary the curve . Vary and to see changes in the plot of the curve, the points in the torsion subgroup that are not the point at infinity, and a geometric illustration of the sum for all .

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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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