Cremona's table of elliptic curves

688 = 2⁴ · 43

Field	Data	Notes
Atkin-Lehner	2- 43+	Signs for the Atkin-Lehner involutions
Class	688b	Isogeny class
Conductor	688	Conductor
∏ cp	2	Product of Tamagawa factors cp
deg	48	Modular degree for the optimal curve
Δ	-11008 = -1 · 28 · 43	Discriminant
Eigenvalues	2-  2  0  4  3 -1 -3 -2	Hecke eigenvalues for primes up to 20
Equation	[0,-1,0,-13,-15]	[a1,a2,a3,a4,a6]
j	-1024000/43	j-invariant
L	2.4922903027455	L(r)(E,1)/r!
Ω	1.2461451513727	Real period
R	1	Regulator
r	0	Rank of the group of rational points
S	1	(Analytic) order of Ш
t	1	Number of elements in the torsion subgroup
Twists	172a1 2752f1 6192n1 17200y1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 688b1

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	4	3	-1	-3	-2	3	6	-5	8	-3	+	12	-9	12	-10	-11	-6	-10	-8	15	0	-1

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Quadratic twists for elliptic curve 688b1

-4	8	-3	5	-7	-11	13	-43
172a1	2752f1	6192n1	17200y1	33712m1	83248bn1	116272n1	29584m1

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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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