Parameterized Families of Elliptic Curves with Large Rational Torsion Subgroups

change curve (t value)

2

zoom

5

order of torsion subgroup

4

recenter plot

show group law lines

[

1

]

{0,0} =

{0,0}

[

2

]

{0,0} =

{-2,0}

[

3

]

{0,0} =

{0,-2}

[

4

]

{0,0} =

O

2

y

+xy+2y

=

3

x

+2

2

x

has a rational point of order

4

at {0,0}

The set of rational points

E()

on an elliptic curve

E/

defined over the rationals



with at least one rational point

O

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

P∈E()

is a rational point of order

n

for

n∈{4,5,6,7,8,9,10,12}

, then

E

is birationally equivalent to an elliptic curve with an equation

E:

2

y

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

3

x

+g(t)

2

x

, where

f(t),q(t)∈(t)

and

(0,0)

is a rational point of order

n

. That is, all elliptic curves

E/

with a rational point of order

n

are in a one-parameter family if

n∈{4,5,6,7,8,9,10,12}

.

In this Demonstration, you can pick from a torsion subgroup of order

n∈{4,5,6,7,8,9,10,12}

and select integer values for the parameter

t

to vary the curve

E:

2

y

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

3

x

+g(t)

2

x

. Vary

t

and

n

to see changes in the plot of the curve, the

n-1

points in the torsion subgroup that are not the point at infinity, and a geometric illustration of the sum

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

for all

1≤k<n

.