A conductor exponent is the exponent with which a prime ideal occurs in a conductor. For example, the Artin
conductor of a Galois representation
has local Artin conductor exponents, while the elliptic
curve conductor 
has local conductor exponents 
.
Conductor exponents measure local ramification. Their precise definition depends on the object whose conductor is being formed; the Artin conductor uses ramification groups, while the elliptic-curve case can be computed using Tate's algorithm.
