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
Δ	-20353792 = -1 · 28 · 433	Discriminant
Eigenvalues	2-  2  0  4  3 -1 -3 -2	Hecke eigenvalues for primes up to 20
Equation	[0,-1,0,67,-79]	[a1,a2,a3,a4,a6]
j	128000000/79507	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	172a2 2752f2 6192n2 17200y2	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 688b2

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 688b2

-4	8	-3	5	-7	-11	13	-43
172a2	2752f2	6192n2	17200y2	33712m2	83248bn2	116272n2	29584m2

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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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