The conductor of an elliptic curve defined over a number field is an ideal of the ring of integers of that is divisible by the prime ideals of bad reduction and no others. It is defined as where the exponent is as follows:
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if has good reduction at ;
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if has multiplicative reduction at ;
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if has additive reduction at and does not lie above either or ; and
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, where is the valuation at , if has additive reduction and lies above or .
For and , 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 of the ideal .
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- Last edited by David Farmer on 2019-09-04 17:25:13
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- dq.ec.source
- ec.conductor_label
- ec.conductor_valuation
- ec.invariants
- ec.q.lmfdb_label
- rcs.source.ec.q
- lmfdb/ecnf/ecnf_stats.py (line 77)
- lmfdb/ecnf/ecnf_stats.py (lines 88-89)
- lmfdb/ecnf/main.py (line 363)
- lmfdb/ecnf/main.py (line 743)
- lmfdb/ecnf/templates/ecnf-curve.html (line 109)
- lmfdb/ecnf/templates/ecnf-curve.html (line 119)
- 2019-09-04 17:25:13 by David Farmer (Reviewed)
- 2019-09-04 17:20:45 by David Farmer
- 2019-08-31 21:39:26 by Andrew Sutherland
- 2019-05-07 12:06:04 by John Cremona
- 2018-06-17 21:50:55 by John Jones (Reviewed)