Cremona's table of elliptic curves

688 = 2⁴ · 43

Field	Data	Notes
Atkin-Lehner	2+ 43+	Signs for the Atkin-Lehner involutions
Class	688a	Isogeny class
Conductor	688	Conductor
∏ cp	1	Product of Tamagawa factors cp
deg	32	Modular degree for the optimal curve
Δ	-11008 = -1 · 28 · 43	Discriminant
Eigenvalues	2+  0 -2  2 -1 -1 -7  6	Hecke eigenvalues for primes up to 20
Equation	[0,0,0,4,-4]	[a1,a2,a3,a4,a6]
Generators	[1:1:1]	Generators of the group modulo torsion
j	27648/43	j-invariant
L	2.0384041307408	L(r)(E,1)/r!
Ω	2.136153901182	Real period
R	0.95424029589479	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	344a1 2752e1 6192e1 17200c1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 688a1

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
+	0	-2	2	-1	-1	-7	6	-9	4	-1	-4	-11	+	0	11	-12	0	-7	10	-4	8	3	6	3

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Quadratic twists for elliptic curve 688a1

-4	8	-3	5	-7	-11	13	-43
344a1	2752e1	6192e1	17200c1	33712c1	83248q1	116272a1	29584e1

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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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