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Computing the rank of elliptic curves over number fields. (English) Zbl 1067.11015
Summary: This paper describes an algorithm of 2-descent for computing the rank of an elliptic curve without 2-torsion, defined over a general number field. This allows one, in practice, to deal with fields of degree from 1 to 5.
The present method is a direct consequence of the arithmetic of invariants of quartics as described J. Cremona [J. Symb. Comput. 31, No. 1-2, 71–87 (2001; Zbl 0965.11025)]. The new ingredients introduced in the present work mainly comprise the solution of Legendre equations over number fields, a method for minimizing some quartics constructed over general number fields, and the implementation of the whole method.

MSC:
11D25 Cubic and quartic Diophantine equations
11R29 Class numbers, class groups, discriminants
11Y50 Computer solution of Diophantine equations
Software:
ecdata
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References:
[1] Simon, Math.Comp (2002)
[2] Smart, The algorithmic resolution of diophantine equations 41 (1998) · Zbl 0907.11001
[3] Cremona, LMS J. Comput. Math 2 pp 62– (1999) · Zbl 0927.11020
[4] Cremona, Algorithms for modular elliptic curves (1997) · Zbl 0872.14041
[5] Cohen, Advanced topics in computational algebraic number theory (2000) · Zbl 0977.11056
[6] Cohen, A course in computational algebraic number theory (1996)
[7] Cassels, Lectures on elliptic curves (1991)
[8] Birch, J. Reine Angew. Math. 212 pp 7– (1963)
[9] DOI: 10.1006/jnth.1996.0006 · Zbl 0859.11034
[10] DOI: 10.1006/jnth.1995.1044 · Zbl 0832.14016
[11] Poonen, J.Reine Angew. Math 488 pp 141– (1997)
[12] Merriman, Acta Arith 11 pp 385– (1996)
[13] Lang, Algebraic number theory 110 (1994)
[14] DOI: 10.1090/S0002-9947-00-02535-6 · Zbl 0954.11022
[15] Djabri, ’A comparison of direct and indirect methods for computing Selmer groups of an elliptic curve’ pp 502– (1998) · Zbl 0915.11034
[16] DOI: 10.2307/2153419 · Zbl 0798.11015
[17] DOI: 10.1090/S0025-5718-99-01055-8 · Zbl 0927.11034
[18] Cremona, Math. Comp (2002)
[19] DOI: 10.4064/aa98-3-4 · Zbl 0972.11058
[20] DOI: 10.1006/jsco.1998.1004 · Zbl 0965.11025
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