Cremona's table of elliptic curves

82 = 2 · 41

Field	Data	Notes
Atkin-Lehner	2+ 41+	Signs for the Atkin-Lehner involutions
Class	82a	Isogeny class
Conductor	82	Conductor
∏ cp	2	Product of Tamagawa factors cp
deg	4	Modular degree for the optimal curve
Δ	164 = 22 · 41	Discriminant
Eigenvalues	2+ -2 -2 -4 -2  4 -2  6	Hecke eigenvalues for primes up to 20
Equation	[1,0,1,-2,0]	[a1,a2,a3,a4,a6]
Generators	[-1:1:1]	Generators of the group modulo torsion
j	389017/164	j-invariant
L	0.58300155947061	L(r)(E,1)/r!
Ω	5.1889950403716	Real period
R	0.22470692491888	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	656c1 2624b1 738i1 2050e1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 82a1

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	-2	-4	-2	4	-2	6	-8	0	-8	2	+	-12	4	-4	8	-14	-2	8	10	4	12	-14	6

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Quadratic twists for elliptic curve 82a1

-4	8	-3	5	-7	-11	13	17	-19	-23	29	-31	37	41
656c1	2624b1	738i1	2050e1	4018f1	9922d1	13858m1	23698d1	29602g1	43378d1	68962k1	78802a1	112258d1	3362b1

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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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