For an elliptic curve 
over 
and a prime 
, the local conductor exponent 
is the exponent of 
in the elliptic curve
conductor. It measures ramification in the local Galois
representation on an 
-adic Tate module of 
, where 
.
The value is 
for good reduction and 
for multiplicative reduction. For additive reduction,
it is 
,
where 
is the contribution from wild ramification.
In particular, 
for 
,
while it may be positive for 
or 3. Ogg's formula gives
 |
where 
is the minimal elliptic discriminant and
is the number of irreducible components of the special fiber of the Néron
minimal model. The exponent can be computed from a local minimal equation in Weierstrass
form using Tate's algorithm.
