Constructing
CM elliptic
curves with
minimal
Galois image
Riccardo
Pengo
7
An explicit example
Consider O = Z[
p
−5], so that ∆O = −20 and HO = Q(
p
−5,i). Then, we can take:
E : y2
= x3
+29736960(36023
p
5−80550)x −55826186240(16154216
p
5+36121925)
so that j(E) = 282880
p
5+632000 and I (E/HO ) = 1. We can take p = 3. Then, 3·O = pp with p = (3,
p
−5+1).
Hence, we have the equality Hp,O = HO , which is reflected by the factorization:
φE,3(x) = 3·(x +594880+59840i −26048
p
−5+266816
p
5)·
(x +594880−59840i +26048
p
−5+266816
p
5)·
(x2
−(1189760+533632
p
5)x −2668089262080−1193205432320
p
5)
of the 3-division polynomial of E. Thus, we have HO (E[p]) = HO (
p
α), where:
α := 13956546560·(1190435+2307955i −1032149
p
−5+532379
p
5)
and E′
= E(α), which gives E′
: y2 −
³
1−i+
p
−5+
p
5
2
´
xy −
³
1+i+
p
−5+
p
5
2
´
y = x3 +x2 +
³
2i −
p
5
´
x −1+2i.