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	682b	Isogeny class
Conductor	682	Conductor
∏ cp	57	Product of Tamagawa factors cp
deg	456	Modular degree for the optimal curve
Δ	-21632647168 = -1 · 219 · 113 · 31	Discriminant
Eigenvalues	2-  0 -2 -3 11- -4  3 -2	Hecke eigenvalues for primes up to 20
Equation	[1,-1,1,359,-6663]	[a1,a2,a3,a4,a6]
Generators	[15:36:1]	Generators of the group modulo torsion
j	5130275528223/21632647168	j-invariant
L	2.6265637235401	L(r)(E,1)/r!
Ω	0.61155331209393	Real period
R	0.075349220442432	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	5456g1 21824b1 6138c1 17050d1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 682b1

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
-	0	-2	-3	-	-4	3	-2	-7	-4	+	7	6	-1	4	-6	9	-6	3	-10	-2	10	-13	6	-7

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Quadratic twists for elliptic curve 682b1

-4	8	-3	5	-7	-11	13	-31
5456g1	21824b1	6138c1	17050d1	33418bl1	7502b1	115258c1	21142b1

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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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