Elliptic curve isogeny class with LMFDB label 136.b (Cremona label 136b)

• Label: 136.b
• Number of curves: $2$
• Conductor: $136$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 136.b have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(17\)	\(1 + T\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 136.b do not have complex multiplication.

Modular form 136.2.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 LMFDB numbering.

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

Isogeny graph

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

Elliptic curves in class 136.b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
136.b1	136b2	\([0, -1, 0, -48, 140]\)	\(6097250/289\)	\(591872\)	\([2]\)	\(16\)	\(-0.13184\)
136.b2	136b1	\([0, -1, 0, -8, -4]\)	\(62500/17\)	\(17408\)	\([2]\)	\(8\)	\(-0.47842\)	\(\Gamma_0(N)\)-optimal