Isogenies - Elliptic Curves: Number Theory and Cryptography - page 384

Therefore, Φ N ( j 1 ,j 2 )=0.

Example 12.1
The curve E 1 : y 2 =4( x 3 − 2 x +1) has j -invariant j 1 = 55296 / 5 and the curve

E 2 : y 2 =4( x 3 − 7 x − 6) has j 2 = 148176 / 25. A calculation (the polynomial

Φ 2 is given on page 329) shows that
Φ 2 55296

=0 ,

, 148176
25 5

so there is a 2-isogeny from E 1 to E 2 . The AGM method (Section 9.4.1)

allows us to compute the period lattices:

L 1 = Z (2 . 01890581997842 ... ) i + Z (2 . 96882494684477 ... )
L 2 = Z (2 . 01890581997842 ... ) i + Z (1 . 48441247342238 ... ) .

The real period for E 1 is twice the real period for E 2 , and the complex periods
are equal. The map C /L 1 → C /L 2 given by z → z gives the 2-isogeny. There

is also a 2-isogeny C /L 2 → C /L 1 given by z → 2 z . We have the factorization

Φ 2 x, 148176
25

= x − x − x − .

132304644
5

55296
5

236276
125

Therefore, E 2 is also isogenous to elliptic curves with j -invariants 132304644 / 5
and 236276 / 125.

We now prove that all nonconstant maps between elliptic curves over C are
linear. This has the interesting consequence that a nonconstant map taking
0to0isoftheform[ α ], hence is a homomorphism.

THEOREM 12.6
Let E 1 = C /L 1 and E 2 = C /L 2 be elliptic curves over C .Suppose hat
f : E 1 → E 2 isananalyticmap(that is, f can be expressed as a pow er series
inaneighborhood of each pointof E 1 ). T hen there exist α, β ∈ C su ch that

f ( z mod L 1 )= αz + β mod L 2

for all z ∈ C .Inparticular, if f (0 mod L 1 )=0mod L 2 and f isnotthe
0-m ap, then f isanisogeny.

f : C → C satisfying

PROOF We can lift f to a continuous map

f ( z mod L 1 )= f ( z )mod L 2

for all z ∈ C (see Exercise 12.13). Moreover, f can be expressed as a power
series in the neighborhood of each point in C (this is the definition of f being