conductor of an elliptic curve

Let E be an elliptic curveMathworldPlanetmath over . For each prime p define the quantity fp as follows:

fp={0, if E has good reduction at p,1, if E has multiplicative reduction at p,2, if E has additive reduction at p, and p2,3,2+δp, if E has additive reduction at p=2or 3.

where δp depends on wild ramification in the action of the inertia group at p of Gal(¯/) on the Tate module Tp(E).


The conductorPlanetmathPlanetmathPlanetmath NE/Q of E/Q is defined to be:


where the product is over all primes and the exponent fp is defined as above.


Let E/:y2+y=x3-x2+2x-2. The primes of bad reduction for E are p=5 and 7. The reduction at p=5 is additive, while the reduction at p=7 is multiplicative. Hence NE/=257=175.


  • 1 James Milne, Elliptic Curves, course notes.
  • 2 Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
  • 3 Joseph H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1994.
Title conductor of an elliptic curve
Canonical name ConductorOfAnEllipticCurve
Date of creation 2013-03-22 13:49:51
Last modified on 2013-03-22 13:49:51
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 9
Author alozano (2414)
Entry type Definition
Classification msc 14H52
Synonym conductor
Related topic EllipticCurve
Related topic LSeriesOfAnEllipticCurve
Related topic ArithmeticOfEllipticCurves
Defines conductor of an elliptic curve