Cremona's table of elliptic curves

689 = 13 · 53

Field	Data	Notes
Atkin-Lehner	13+ 53+	Signs for the Atkin-Lehner involutions
Class	689a	Isogeny class
Conductor	689	Conductor
∏ cp	1	Product of Tamagawa factors cp
deg	40	Modular degree for the optimal curve
Δ	689 = 13 · 53	Discriminant
Eigenvalues	-1 -2 -2  2  2 13+ -2  4	Hecke eigenvalues for primes up to 20
Equation	[1,0,0,-14,19]	[a1,a2,a3,a4,a6]
Generators	[1:2:1]	Generators of the group modulo torsion
j	304821217/689	j-invariant
L	1.033304817561	L(r)(E,1)/r!
Ω	5.104722329004	Real period
R	0.80968542534816	Regulator
r	1	Rank of the group of rational points
S	1	(Analytic) order of Ш
t	2	Number of elements in the torsion subgroup
Twists	11024f1 44096j1 6201d1 17225e1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 689a1

a2	a3	a5	a7	a11	a13	a17	a19	a23	a29	a31	a37	a41	a43	a47	a53	a59	a61	a67	a71	a73	a79	a83	a89	a97
-1	-2	-2	2	2	+	-2	4	6	6	-8	-6	-10	-12	6	+	-6	10	-8	-12	14	-10	12	10	-14

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Quadratic twists for elliptic curve 689a1

-4	8	-3	5	-7	-11	13	53
11024f1	44096j1	6201d1	17225e1	33761c1	83369f1	8957a1	36517a1

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Data from Elliptic Curve Data by J. E. Cremona.
Design inspired by The Modular Forms Explorer by William Stein.

Part of Computational Number Theory
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