Elliptic curve isogeny class with Cremona label 37b (LMFDB label 37.b)

• Label: 37b
• Number of curves: $3$
• Conductor: $37$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 37b have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(37\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(2\)	\( 1 + 2 T^{2}\)	1.2.a
\(3\)	\( 1 - T + 3 T^{2}\)	1.3.ab
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 - 3 T + 11 T^{2}\)	1.11.ad
\(13\)	\( 1 + 4 T + 13 T^{2}\)	1.13.e
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(19\)	\( 1 - 2 T + 19 T^{2}\)	1.19.ac
\(23\)	\( 1 - 6 T + 23 T^{2}\)	1.23.ag
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 37b do not have complex multiplication.

Modular form 37.2.a.b

Isogeny matrix

The \((i,j)\)-th 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.

\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 37b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
37.b2	37b1	\([0, 1, 1, -23, -50]\)	\(1404928000/50653\)	\(50653\)	\([3]\)	\(2\)	\(-0.32722\)	\(\Gamma_0(N)\)-optimal
37.b1	37b2	\([0, 1, 1, -1873, -31833]\)	\(727057727488000/37\)	\(37\)	\([]\)	\(6\)	\(0.22208\)
37.b3	37b3	\([0, 1, 1, -3, 1]\)	\(4096000/37\)	\(37\)	\([3]\)	\(6\)	\(-0.87653\)