For an elliptic curve 
over 
, the conductor measures its bad reduction and the ramification
of its torsion points. It is the positive integer
 |
(1)
|
where the product is over primes and 
is the local conductor
exponent. The primes dividing 
are exactly the primes at which 
has bad reduction.
The local exponents are
 |
(2)
|
At the primes 2 and 3, additive reduction can include an additional contribution from wild ramification. In this case, 
and 
. The exponents, together with the reduction
type and a local minimal equation in Weierstrass
form, can be computed by Tate's algorithm.
The conductor is invariant under isogeny over 
.
An elliptic curve over 
is semistable iff its conductor
is squarefree. By the modularity theorem,
is also the modular form level of the weight-two
modular newform associated with 
, and it occurs in the functional
equation of the Hasse-Weil L-function
of 
.
."