# Parameterized Families of Elliptic Curves with Large Rational Torsion Subgroups

Parameterized Families of Elliptic Curves with Large Rational Torsion Subgroups

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 .

E()

E/

O

P∈E()

n

n∈{4,5,6,7,8,9,10,12}

E

E:+f(t)xy+g(t)y=+g(t)

2

y

3

x

2

x

f(t),q(t)∈(t)

(0,0)

n

E/

n

n∈{4,5,6,7,8,9,10,12}

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 .

n∈{4,5,6,7,8,9,10,12}

t

E:+f(t)xy+g(t)y=+g(t)

2

y

3

x

2

x

t

n

n-1

(0,0)+[k](0,0)

1≤k<n