Complex Multiplication - Elliptic Curves: Number Theory and Cryptography - page 321

10.3 Integrality of j-invariants

At the end of Section 10.1, we showed that the j -invariant of a lattice,
or of a complex elliptic curve, with complex multiplication by an order in an
imaginary quadratic field is algebraic over Q . This means that the j -invariant
is a root of a polynomial with rational coecients. In the present section, we
show that this j -invariant is an algebraic integer, so it is a root of a monic
polynomial with integer coecients.

THEOREM 10.9
Let R be an order inanimaginary quadraticfieldand et L be a lattice w ith
RL L .Then j ( L ) isanalgebraicinteger. E quivalently, let E be an elliptic

curve over C with com plex m ultiplication. T hen j ( E ) isanalgebraicinteger.

⊆

The proof of the theorem will occupy the remainder of th is se ction. The

theorem has an amusing consequence. The ring R = Z 1+ √ − 16 2 is a prin-
cipal ideal domain (see [16]), so there is only one equivalence class of ideals
of R , namely the one represented by R . The proof of Proposition 10.4 shows
that all automorphisms of C must fix j ( R ), where R is regarded as a lattice.
Therefore, j ( R )

∈ Q . The only algebraic integers in Q are the elements of Z ,

so j ( R ) ∈ Z . Recall that j ( τ )isthe j -invariant of the lattice Z τ + Z ,andthat

j ( τ )= 1

q + 744 + 196884 q + 21493760 q 2 +

··· ,

where q = e 2 πiτ .When τ = 1+ √ − 163
2

,wehave R = Z τ + Z and

q = −e −π √ 163 .

Therefore,

e π √ 163 + 744 196884 e −π √ 163 + 21493760 e − 2 π √ 163 + − − ···∈ Z .

Since

196884 e −π √ 163 21493760 e − 2 π √ 163 + < 10 − 12 ,

we find that e π √ 163 differs from an integer by less than 10 − 12 .Infact,

− ···

e π √ 163 = 262537412640768743 . 999999999999250 ...,

as predicted. In the days when high precision ca lcu lation was not widely
available, it was often claimed as a joke that e π √ 163 was an integer. Any
calculation with up to 30 places of accuracy seemed to indicate that this was