Elliptic curve isogeny class with Cremona label 6137b (LMFDB label 6137.b)

• Label: 6137b
• Number of curves: $4$
• Conductor: $6137$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 6137b have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(17\)	\(1 - T\)
\(19\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 - T + 2 T^{2}\)	1.2.ab
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 6137b do not have complex multiplication.

Modular form 6137.2.a.b

Isogeny matrix

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 6137b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
6137.b3	6137b1	\([1, -1, 0, -248, -581]\)	\(35937/17\)	\(799779977\)	\([2]\)	\(1728\)	\(0.40244\)	\(\Gamma_0(N)\)-optimal
6137.b2	6137b2	\([1, -1, 0, -2053, 35880]\)	\(20346417/289\)	\(13596259609\)	\([2, 2]\)	\(3456\)	\(0.74901\)
6137.b1	6137b3	\([1, -1, 0, -32738, 2288159]\)	\(82483294977/17\)	\(799779977\)	\([2]\)	\(6912\)	\(1.0956\)
6137.b4	6137b4	\([1, -1, 0, -248, 95445]\)	\(-35937/83521\)	\(-3929319027001\)	\([2]\)	\(6912\)	\(1.0956\)