Elliptic curve isogeny class with LMFDB label 12138.bb (Cremona label 12138ba)

• Label: 12138.bb
• Number of curves: $4$
• Conductor: $12138$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 12138.bb have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1 - T\)
\(7\)	\(1 - T\)
\(17\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 - 8 T + 23 T^{2}\)	1.23.ai
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 12138.bb do not have complex multiplication.

Modular form 12138.2.a.bb

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 LMFDB numbering.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 12138.bb

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
12138.bb1	12138ba4	\([1, 0, 0, -1467837, 684363897]\)	\(14489843500598257/6246072\)	\(150764993878968\)	\([2]\)	\(221184\)	\(2.0632\)
12138.bb2	12138ba3	\([1, 0, 0, -196237, -17702647]\)	\(34623662831857/14438442312\)	\(348508897558419528\)	\([2]\)	\(221184\)	\(2.0632\)
12138.bb3	12138ba2	\([1, 0, 0, -92197, 10575425]\)	\(3590714269297/73410624\)	\(1771954002133056\)	\([2, 2]\)	\(110592\)	\(1.7166\)
12138.bb4	12138ba1	\([1, 0, 0, 283, 495105]\)	\(103823/4386816\)	\(-105887073890304\)	\([2]\)	\(55296\)	\(1.3700\)	\(\Gamma_0(N)\)-optimal