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Number of elliptic curves (up to isomorphism) of conductor n.
3

%I #9 Nov 05 2024 12:16:46

%S 0,0,0,0,0,0,0,0,0,0,3,0,0,6,8,0,4,0,3,4,6,0,0,6,0,5,4,0,0,8,0,4,4,4,

%T 3,4,4,5,4,4,0,6,1,2,8,2,0,6,4,8,2,2,1,6,4,6,7,3,0,0,1,4,6,4,2,12,1,0,

%U 2,4,0,6,2,0,12,1,6,4,1,8,0,2,1,6,2,0,0,1,3,16,4,3,0,2,0,8,0,6,11,4,1,12,0

%N Number of elliptic curves (up to isomorphism) of conductor n.

%H J. E. Cremona, <a href="/A110620/b110620.txt">Table of n, a(n) for n = 1..10000</a>

%H A. Brumer and J. H. Silverman, <a href="https://doi.org/10.1007/BF02567942">The number of elliptic curves over Q with conductor N</a>, Manuscripta Math. 91 (1996), no. 1, 95-102.

%H J. E. Cremona, <a href="https://johncremona.github.io/ecdata/">Elliptic Curve Data</a>.

%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/">Elliptic curves over Q</a>.

%e a(11)=3 since there are three non-isomorphic elliptic curves of conductor eleven, represented by the minimal models y^2+y=x^3-x^2-10*x-20, y^2+y=x^3-x^2-7820*x-263580 and y^2+y=x^3-x^2.

%o (Sage) # Uses Cremona's database of elliptic curves (works for all n < 500000)

%o def a(n):

%o return CremonaDatabase().number_of_curves(n) # _Robin Visser_, Nov 04 2024

%Y Cf. A005788, A060564.

%K nonn,changed

%O 1,11

%A _Steven Finch_, Sep 14 2005