Cremona's table of elliptic curves

682 = 2 · 11 · 31

Field	Data	Notes
Atkin-Lehner	2- 11+ 31-	Signs for the Atkin-Lehner involutions
Class	682a	Isogeny class
Conductor	682	Conductor
∏ cp	1	Product of Tamagawa factors cp
Δ	-146192756842 = -1 · 2 · 119 · 31	Discriminant
Eigenvalues	2- -2  0 -1 11+ -4 -3  2	Hecke eigenvalues for primes up to 20
Equation	[1,0,0,-2003,-39269]	[a1,a2,a3,a4,a6]
Generators	[1054:10753:8]	Generators of the group modulo torsion
j	-888751018248625/146192756842	j-invariant
L	2.2812508095831	L(r)(E,1)/r!
Ω	0.35363003593532	Real period
R	6.4509531933548	Regulator
r	1	Rank of the group of rational points
S	1	(Analytic) order of Ш
t	1	Number of elements in the torsion subgroup
Twists	5456h3 21824m3 6138i3 17050b3	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 682a3

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
-	-2	0	-1	+	-4	-3	2	-3	-6	-	-7	0	-1	-6	-6	3	8	5	12	2	-10	15	0	17

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Quadratic twists for elliptic curve 682a3

-4	8	-3	5	-7	-11	13	-31
5456h3	21824m3	6138i3	17050b3	33418be3	7502c3	115258f3	21142d3

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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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