Conductor

The conductor of an elliptic curve E over a field K is an integer, (or an “ideal” of the ring of integers in any algebraic field,) defined by its prime factorization:

$N=\prod _{\mathrm {all\;primes\;} p}p^{f_{p}}$

where the exponent f_p of each prime p is given by

$f_{p}=0$ if E has “good” reduction at p;
$f_{p}=1$ if E has “multiplicative” reduction at p;
$f_{p}=2$ if E has “additive” reduction at p > 3;
$f_{p}=2+\delta _{p}$ if E has “additive” reduction at p = 2 or 3;

where $\delta _{p}$ is some measure of “wild ramification” in the action of the inertia group on $T_{\ell }(E)$ for which Silverman refers to Serre and Tate and Ogg who in turn rely on the minimal models of Néron for a precise definition.

Only a finite number of “bad” reductions are possible on an elliptic curve E.

This all needs to be simplified and brought down from any excessive abstraction. Cryptography is a martial art. References are inconsistent, disinformation exists, and it is unwise to accept mathematical results on face value which one does not prove oneself, or work through and satisfy oneself with the proofs. If anything is too difficult, another tack needs to be taken, a plan of attack from a different position. It is that strange feeling of invading a foreign library and opening up books one is not really welcome to.