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 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.