Elliptic Curves over C - Elliptic Curves: Number Theory and Cryptography - page 301

and ψ n ( X ) has degree n 2 − 1. Therefore,

P ( X )= X n 2 + ··· ,

so x is of degree at most n 2 over K ( g n ( x )). Since

[ K ( x, y ): K ( x )] = 2 ,

we obtain

[ K ( x, y ): K ( g n ( x ))] ≤ 2 n 2 .

Combined with the previous inequality from above, we obtain equality, which
means that we had equality in all of our calculations. In particular,

F = K ( g n ( x ) ,yh n ( x )) .

The functions in F are those that are invariant under translation by elements
of E [ n ]. Those on the right are those that are of the form h ( n ( x, y )). There-
fore, we have proved the proposition.

9.6 The Torsion Subgroup: Doud's Method
Let E : y 2 = x 3 + Ax + B be an elliptic curve defined over Z . The Lutz-
Nagell Theorem (Section 8.1) says that if ( x, y )

∈ E ( Q ) is a torsion point,

then either y =0or y 2 4 A 3 +27 B 2 . This allows us to determine the torsion,

as long as we can factor 4 A 3 +27 B 2 , and as long as it does not have many
square factors. In this section, we present an algorithm due to Doud [35] that
avoids these di culties and is usually much faster in practice.
Let p ≥ 11 be a prime not dividing 4 A 3 +27 B 2 . By Theorem 8.9, the kernel
of the map from the torsion of E ( Q )to E ( F p ) is trivial. Therefore, the order
of the torsion subgroup of E ( Q ) divides # E ( F p ). If we use a few values of
p and take the greatest common divisor of the values of # E ( F p ), then we
obtain a value b that is a multiple of the order of the torsion subgroup of
E ( Q ). We consider divisors n of b , running from largest divisor to smallest,
and look for a point of order n on E (of course, we should look at only the
values of n allowed by Mazur's theorem).
In order to work analytically, we multiply the equation for E by 4 to obtain
E 1 : y 1 =4 x 3 +4 Ax +4 B ,with y 1 =2 y .
The period lattice for E 1 is generated by ω 1 and ω 2 ,with ω 2 ∈ R .The
points in the fundamental parallelogram corresponding to real x, y under the
map of Theorem 9.10 lie on the line ω 2 R , and also on the line

|

1

2 ω 1 + ω 2 R
when the cubic polynomial 4 x 3 +4 Ax +4 B has 3 real roots. Doubling a point
on the second line yields a point on the first line. Therefore, if n is odd, all