Cremona's table of elliptic curves

2139 = 3 · 23 · 31

Field	Data	Notes
Atkin-Lehner	3+ 23+ 31+	Signs for the Atkin-Lehner involutions
Class	2139b	Isogeny class
Conductor	2139	Conductor
∏ cp	1	Product of Tamagawa factors cp
deg	3360	Modular degree for the optimal curve
Δ	436364746371 = 37 · 235 · 31	Discriminant
Eigenvalues	1 3+  3  3 -3 -6 -2 -4	Hecke eigenvalues for primes up to 20
Equation	[1,1,0,-2761,44782]	[a1,a2,a3,a4,a6]
Generators	[-22:320:1]	Generators of the group modulo torsion
j	2328995685476377/436364746371	j-invariant
L	3.7679396878677	L(r)(E,1)/r!
Ω	0.89428180173396	Real period
R	4.2133695224055	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	34224bo1 6417j1 53475m1 104811s1	Quadratic twists by: -4 -3 5 -7

Hecke Eigenvalues for elliptic curve 2139b1

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
1	+	3	3	-3	-6	-2	-4	+	-9	+	1	0	-8	7	0	-3	13	-7	9	-4	4	-7	-12	-6

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Quadratic twists for elliptic curve 2139b1

-4	-3	5	-7	-23	-31
34224bo1	6417j1	53475m1	104811s1	49197f1	66309m1

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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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