Cremona's table of elliptic curves

684 = 2² · 3² · 19

Field	Data	Notes
Atkin-Lehner	2- 3- 19+	Signs for the Atkin-Lehner involutions
Class	684a	Isogeny class
Conductor	684	Conductor
∏ cp	6	Product of Tamagawa factors cp
deg	144	Modular degree for the optimal curve
Δ	-3545856 = -1 · 28 · 36 · 19	Discriminant
Eigenvalues	2- 3-  1 -3 -5 -4  3 19+	Hecke eigenvalues for primes up to 20
Equation	[0,0,0,-192,1028]	[a1,a2,a3,a4,a6]
Generators	[4:18:1]	Generators of the group modulo torsion
j	-4194304/19	j-invariant
L	2.1232510163787	L(r)(E,1)/r!
Ω	2.5117117385399	Real period
R	0.14089004083015	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	2736r1 10944bd1 76a1 17100r1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 684a1

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	-3	-5	-4	3	+	-8	2	4	10	-10	1	1	4	-6	-13	-12	-2	9	8	12	-12	-8

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Quadratic twists for elliptic curve 684a1

-4	8	-3	5	-7	-11	13	-19
2736r1	10944bd1	76a1	17100r1	33516u1	82764m1	115596v1	12996n1

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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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