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), 3352. Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975.
 J.H. Silverman, Advanced topics in the arithmetic of elliptic curves, GTM 151, SpringerVerlag, New York, 1994.
The conductor norm is the norm $[\mathcal{O}_K:\mathfrak{n}]$ of the ideal $\mathfrak{n}$.
 Review status: reviewed
 Last edited by David Farmer on 20190904 17:25:13
 20190904 17:25:13 by David Farmer (Reviewed)
 20190904 17:20:45 by David Farmer
 20190831 21:39:26 by Andrew Sutherland
 20190507 12:06:04 by John Cremona
 20180617 21:50:55 by John Jones (Reviewed)