Cremona's table of elliptic curves

682 = 2 · 11 · 31

Field	Data	Notes
Atkin-Lehner	2- 11+ 31-	Signs for the Atkin-Lehner involutions
Class	682a	Isogeny class
Conductor	682	Conductor
∏ cp	9	Product of Tamagawa factors cp
deg	72	Modular degree for the optimal curve
Δ	-174592 = -1 · 29 · 11 · 31	Discriminant
Eigenvalues	2- -2  0 -1 11+ -4 -3  2	Hecke eigenvalues for primes up to 20
Equation	[1,0,0,-33,73]	[a1,a2,a3,a4,a6]
Generators	[-6:11:1]	Generators of the group modulo torsion
j	-3981876625/174592	j-invariant
L	2.2812508095831	L(r)(E,1)/r!
Ω	3.1826703234178	Real period
R	0.71677257703942	Regulator
r	1	Rank of the group of rational points
S	1	(Analytic) order of Ш
t	3	Number of elements in the torsion subgroup
Twists	5456h1 21824m1 6138i1 17050b1	Quadratic twists by: -4 8 -3 5

Hecke Eigenvalues for elliptic curve 682a1

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	-1	+	-4	-3	2	-3	-6	-	-7	0	-1	-6	-6	3	8	5	12	2	-10	15	0	17

---

Quadratic twists for elliptic curve 682a1

-4	8	-3	5	-7	-11	13	-31
5456h1	21824m1	6138i1	17050b1	33418be1	7502c1	115258f1	21142d1

---

Data from Elliptic Curve Data by J. E. Cremona.
Design inspired by The Modular Forms Explorer by William Stein.

Part of Computational Number Theory
Back to Tables and computations