Modular Elliptic Curves over Q(5) - William Stein

Curves over
Q( √ 5)
W. Stein
Background
The Plan
Conductor
What is the “Conductor” of an Elliptic Curve?
If E, given by y 2 + a1xy + a3y = x 3 + a2x 2 + a4x + a6 is a
Weierstrass equation for an elliptic curve over F , it has a
discriminant, which is a polynomial in the ai:
∆ = −a 4 1 a2a 2 3 + a5 1 a3a4 − a 6 1 a6 − 8a 2 1 a2 2 a2 3 + a3 1 a3 3 + 8a3 1 a2a3a4 + a 4 1 a2 4 − 12a4 1 a2a6 − 16a 3 2 a2 3 +
36a1a2a 3 3 + 16a1a 2 2 a3a4 − 30a 2 1 a2 3 a4 + 8a 2 1 a2a 2 4 − 48a2 1 a2 2 a6 + 36a 3 1 a3a6 − 27a 4 3 + 72a2a 2 3 a4 + 16a 2 2 a2 4 −
96a1a3a 2 4 − 64a3 2 a6 + 144a1a2a3a6 + 72a 2 1 a4a6 − 64a 3 4 − 216a2 3 a6 + 288a2a4a6 − 432a 2 6
Among all models for E, there are some that simultaneously minimize
ordp(∆) for all primes p: the minimal discriminant.
Reducing such a model modulo p will give either an elliptic curve, or
a nodal cubic (e.g., y 2 = x 2 (x − 1)), or a cuspidal cubic (e.g.,
y 2 = x 3 ). The conductor of E is the ideal
n =
p fp
p|∆
where fp = 1 if reduction is nodal, and fp ≥ 2 if cuspidal...