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:

[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.