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	684b	Isogeny class
Conductor	684	Conductor
∏ cp	24	Product of Tamagawa factors cp
deg	144	Modular degree for the optimal curve
Δ	-113689008 = -1 · 24 · 39 · 192	Discriminant
Eigenvalues	2- 3- -2  0 -2  2 -6 19+	Hecke eigenvalues for primes up to 20
Equation	[0,0,0,24,-511]	[a1,a2,a3,a4,a6]
Generators	[10:27:1]	Generators of the group modulo torsion
j	131072/9747	j-invariant
L	2.0174937497296	L(r)(E,1)/r!
Ω	0.89143385110214	Real period
R	0.37720012300694	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	2736u1 10944bf1 228a1 17100m1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 684b1

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	-2	2	-6	+	-2	-4	-8	-2	8	-8	-2	4	0	2	12	4	6	-16	-6	0	-2

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Quadratic twists for elliptic curve 684b1

-4	8	-3	5	-7	-11	13	-19
2736u1	10944bf1	228a1	17100m1	33516v1	82764n1	115596w1	12996o1

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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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