Conductor of an elliptic curve (reviewed)

The conductor of an elliptic curve $E$ defined over a number field $K$ is an ideal of the ring of integers of $K$ that is divisible by the prime ideals of bad reduction and no others. It is defined as $$ \mathfrak{n} = \prod_{\mathfrak{p}}\mathfrak{p}^{e_{\mathfrak{p}}} $$ where the exponent $e_{\mathfrak{p}}$ is as follows:

• $e_{\mathfrak{p}}=0$ if $E$ has good reduction at $\mathfrak{p}$;

• $e_{\mathfrak{p}}=1$ if $E$ has multiplicative reduction at $\mathfrak{p}$;

• $e_{\mathfrak{p}}=2$ if $E$ has additive reduction at $\mathfrak{p}$ and $\mathfrak{p}$ does not lie above either $2$ or $3$; and

• $2\leq e_{\mathfrak{p}}\leq 2+6v_{\mathfrak{p}}(2)+3v_{\mathfrak{p}}(3)$, where $v_{\mathfrak{p}}$ is the valuation at $\mathfrak{p}$, if $E$ has additive reduction and $\mathfrak{p}$ lies above $2$ or $3$.

For $\mathfrak{p}=2$ and $3$, there is an algorithm of Tate that simultaneously creates a 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.
• J.H. Silverman, Advanced topics in the arithmetic of elliptic curves, GTM 151, Springer-Verlag, New York, 1994.

The conductor norm is the norm $[\mathcal{O}_K:\mathfrak{n}]$ of the ideal $\mathfrak{n}$.