Conductor of an elliptic curve over $\Q$ (reviewed)

The conductor $N$ of an elliptic curve $E$ defined over $\Q$ is a positive integer divisible by the primes of bad reduction and no others. It has the form $N=\prod p^{e_p}$, where the exponent $e_p$ is

• $e_p=1$ if $E$ has multiplicative reduction at $p$,

• $e_p=2$ if $E$ has additive reduction at $p$ and $p\ge5$,

• $2\leq e_p\leq 5$ if $E$ has additive reduction and $p=3$, and

• $2\leq e_p\leq 8$ if $E$ has additive reduction and $p=2$.

For all primes $p$, there is an algorithm of Tate that simultaneously creates a local minimal Weierstrass equation and computes the exponent of the conductor. See:

• J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil, Modular functions of one variable, IV (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), 33-52. Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975. [MR:0393039]
• J.H. Silverman, Advanced topics in the arithmetic of elliptic curves, GTM 151, Springer-Verlag, New York, 1994.[MR:1312368]