cp's OEIS Frontend

A110563 Conductors of positive-rank elliptic curves.

%I A110563 #23 Nov 11 2024 08:59:15
%S A110563 37,43,53,57,58,61,65,77,79,82,83,88,89,91,92,99,101,102,106,112,117,
%T A110563 118,121,122,123,124,128,129,130,131,135,136,138,141,142,143,145,148,
%U A110563 152,153,154,155,156,158,160,162,163,166,170,171,172,175,176,184,185,189,190,192,196,197,198
%N A110563 Conductors of positive-rank elliptic curves.
%H A110563 J. E. Cremona, <a href="/A110563/b110563.txt">Table of n, a(n) for n = 1..10000</a>
%H A110563 J. E. Cremona, <a href="https://johncremona.github.io/ecdata/">Elliptic Curve Data</a>
%H A110563 Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/">Elliptic curves over Q</a> [Broken link]
%H A110563 Steven R. Finch, <a href="https://citeseerx.ist.psu.edu/pdf/08dee3afaf3c76273bc560f4c91ec11e5d18cc30">Elliptic curves over Q</a>
%H A110563 LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/">Elliptic curves over Q</a>.
%H A110563 Heinz M. Tschoepe and Horst G. Zimmer, <a href="https://doi.org/10.1090/S0025-5718-1987-0866121-6">Computation of the Néron-Tate height on elliptic curves</a>, Math. Comp. 48 (1987) 351-370.
%e A110563 a(1) = 37, as there are no positive rank elliptic curves over Q of conductor less than 37, but there is an elliptic curve of rank 1 over Q of conductor equal to 37, given by E : y^2 + y = x^3 - x. - _Robin Visser_, Nov 07 2024
%o A110563 (Sage) # Uses Cremona's database of elliptic curves (works for all k < 500000)
%o A110563 def is_A110563(k):
%o A110563     curves = [EllipticCurve(i[0]) for i in CremonaDatabase().allcurves(k).values()]
%o A110563     return any([(E.rank() > 0) for E in curves])
%o A110563 print([k for k in range(1, 100) if is_A110563(k)])  # _Robin Visser_, Nov 07 2024
%Y A110563 Cf. A005788.
%K A110563 nonn
%O A110563 1,1
%A A110563 _Steven Finch_, Sep 12 2005
%E A110563 More terms added by _Robin Visser_, Nov 07 2024, taken from J. E. Cremona's database of elliptic curves.