A110563 Conductors of positive-rank elliptic curves.

37, 43, 53, 57, 58, 61, 65, 77, 79, 82, 83, 88, 89, 91, 92, 99, 101, 102, 106, 112, 117, 118, 121, 122, 123, 124, 128, 129, 130, 131, 135, 136, 138, 141, 142, 143, 145, 148, 152, 153, 154, 155, 156, 158, 160, 162, 163, 166, 170, 171, 172, 175, 176, 184, 185, 189, 190, 192, 196, 197, 198

Offset: 1

Author

Steven Finch, Sep 12 2005

Keywords

• nonn

Examples

			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
		

Links

• J. E. Cremona, Table of n, a(n) for n = 1..10000
• J. E. Cremona, Elliptic Curve Data
• Steven R. Finch, Elliptic curves over Q [Broken link]
• Steven R. Finch, Elliptic curves over Q
• LMFDB, Elliptic curves over Q.
• Heinz M. Tschoepe and Horst G. Zimmer, Computation of the Néron-Tate height on elliptic curves, Math. Comp. 48 (1987) 351-370.

Crossrefs

Cf. A005788.

Programs

Sage

# Uses Cremona's database of elliptic curves (works for all k < 500000)
def is_A110563(k):
    curves = [EllipticCurve(i[0]) for i in CremonaDatabase().allcurves(k).values()]
    return any([(E.rank() > 0) for E in curves])
print([k for k in range(1, 100) if is_A110563(k)])  # Robin Visser, Nov 07 2024

Extensions

More terms added by Robin Visser, Nov 07 2024, taken from J. E. Cremona's database of elliptic curves.