Cremona's table of elliptic curves

688 = 2⁴ · 43

Field	Data	Notes
Atkin-Lehner	2- 43-	Signs for the Atkin-Lehner involutions
Class	688c	Isogeny class
Conductor	688	Conductor
∏ cp	1	Product of Tamagawa factors cp
deg	80	Modular degree for the optimal curve
Δ	-176128 = -1 · 212 · 43	Discriminant
Eigenvalues	2-  2 -4  0 -3 -5 -3  2	Hecke eigenvalues for primes up to 20
Equation	[0,-1,0,-5,-19]	[a1,a2,a3,a4,a6]
Generators	[4:3:1]	Generators of the group modulo torsion
j	-4096/43	j-invariant
L	2.3637294195545	L(r)(E,1)/r!
Ω	1.3631824181704	Real period
R	1.7339788043386	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	43a1 2752d1 6192ba1 17200o1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 688c1

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

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Quadratic twists for elliptic curve 688c1

-4	8	-3	5	-7	-11	13	-43
43a1	2752d1	6192ba1	17200o1	33712r1	83248bh1	116272q1	29584n1

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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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