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 = 2 or 3;
$f_{p}=2+???$ if E has “additive” reduction at p > 3;

N.B.: The definition here is incomplete and more authoritative references on the advanced abstract algebra should be consulted. All types of reduction at each prime p that are not “good” are said to be “bad,” and there are several special cases; only a finite number of “bad” reductions are possible.

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.