Isomorphic elliptic curves

Elliptic curves with the same j-invariant are isomorphic. But exactly where are they isomorphic?

Consider the elliptic curve over the rationals given by

y2 = x3 + b

Let's use Etors to represent the group of points of finite order on the curve E and #Etors to represent the number of points in Etors. The structure of Etors as well as the value of #Etors of depends on what the value of b is. Here's what we get for a few different curves with different values of b:

Curve

#Etors

Structure of Etors

y2 = x3 + 1

6

Z6 = <(2,3)>

y2 = x3 + 2

1

Z1 ={O}

y2 = x3 + 4

3

Z3 = <(0,2)>

y2 = x3 + 8

2

Z2 = <(-2,0)>

y2 = x3 + 9

3

Z3 = <(0,3)>

Each of these curves has the same j-invariant. In each case we have that j = 0, but the structure of Etors varies from curve to curve.

Here's another example of curves with the same j-invariant that have different structures for Etors:

Curve

#Etors

Structure of Etors

y2 = x3 + x

2

Z2 = <(0,0)>

y2 = x3 + 4x

4

Z4 = <(2,4)>

y2 = x3 – 4x

4

Z2 × Z2 = <(2,0),(0,0)>

Curves with the same j-invariant are supposed to be isomorphic. What's going on here?

Curves with the same j-invariant are only isomorphic over some extension field, not over the field that the elliptic curves are defined over. So although the curves in these examples aren't isomorphic over the rational numbers, they're isomorphic over some extension to the rational numbers.

Let E be the elliptic curve given by

y2 = x3 +1

and E′ be the elliptic curve given by

y2 = x3 + 4

We can write the isomorphism φ:EE′ as

φ(x,y) = (c2x,c3y)

where

c = ∛2

Here's what we get when we look at what happens to the subgroup of points on E generated by P = (2,3) under the isomorphism φ, where φ(P) = P′, etc.

Point on E

Multiple of P

Point on E

Multiple of P

(2,3)

1

(25/3,6)

1

(0,1)

2

(0,2)

2

(-1,0)

3

(-22/3,0)

3

(0,-1)

4

(0,-2)

4

(2,-3)

5

(25/3,-6)

5

O

6

O

6

Not all of the multiples of  φ(P) = P′ have rational coordinates, but the ones that do give us a subgroup isomorphic to Z3. So if we insist on rational coordinates, then we find that we only have three points of finite order, but if we extend what we allow for the coordinates of points to include ∛2 then we find that we have all six points. So although E and E′ aren't isomorphic over the rationals, they're isomorphic over an extension of the rationals that includes ∛2.

The "tors" in "Etors" stands for "torsion." Points of finite order on an elliptic curve are sometimes called "torsion points," but nobody quite seems to know exactly why. If you know this bit of elliptic curve history, be sure to let me know.

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