Notions of
Level, Conductor (technical)

•Given an elliptic curve
E over **F**q, let End(E)
denote the endomorphisms of E

( = isogenies + trivial, zero map)

which
are defined over the algebraic
closure of **F**q.

•For an *ordinary* elliptic curve, End(E) is
an order in some imaginary quadratic number field **K = ****Q****(****p****-d).**

•This field K is an invariant of the
isogeny class

(called the “**C**omplex
**M**ultiplication Field”)

(called the “

•Orders are always of the
form **O****D**** = ****Z****+c****O****K**, where **O****K** is the ring of algebraic integers in K
(solutions to monic integral polynomials).

•The discriminant of the
order **O****D** is related to the discriminant d of K by **D=c****2****d**. Curves for a given constant value of c form
levels.

•Isogenies can therefore be of two forms:

–They can preserve D (“horizontal”).

–Or they can change D (“vertical”).

•Supersingular curves all lie on the same level (by
definition), so this is really an issue pertaining to ordinary curves.