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

  • ep=0e_{\mathfrak{p}}=0 if EE has good reduction at p\mathfrak{p};

  • ep=1e_{\mathfrak{p}}=1 if EE has multiplicative reduction at p\mathfrak{p};

  • ep=2e_{\mathfrak{p}}=2 if EE has additive reduction at p\mathfrak{p} and p\mathfrak{p} does not lie above either 22 or 33; and

  • 2ep2+6vp(2)+3vp(3)2\leq e_{\mathfrak{p}}\leq 2+6v_{\mathfrak{p}}(2)+3v_{\mathfrak{p}}(3), where vpv_{\mathfrak{p}} is the valuation at p\mathfrak{p}, if EE has additive reduction and p\mathfrak{p} lies above 22 or 33.

For p=2\mathfrak{p}=2 and 33, 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 [OK:n][\mathcal{O}_K:\mathfrak{n}] of the ideal n\mathfrak{n}.

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  • Last edited by David Farmer on 2019-09-04 17:25:13
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