It is given as a product of prime ideals, together with associated exponents, which encode the ramification in the field extensions generated by the points of finite order in the group law of the elliptic curve.
Ogg's formula expresses the conductor in terms of the discriminant and the number of components of the special fiber over a local field, which can be computed using Tate's algorithm.
He showed that it could be written as a product over primes with exponents given by order(Δ) − μ + 1, which by Ogg's formula is equal to ε+δ.
We can write the exponent f of the conductor as a sum ε + δ of two terms, corresponding to the tame and wild ramification.
The wild ramification term δ is zero unless p divides 2 or 3, and in the latter cases it is defined in terms of the wild ramification of the extensions of K by the division points of E by Serre's formula Here M is the group of points on the elliptic curve of order l for a prime l, P is the Swan representation, and G the Galois group of a finite extension of K such that the points of M are defined over it (so that G acts on M) The exponent of the conductor is related to other invariants of the elliptic curve by Ogg's formula: where n is the number of components (without counting multiplicities) of the singular fibre of the Néron minimal model for E. (This is sometimes used as a definition of the conductor).