Elliptic curve isogeny class with Cremona label 81312bv (LMFDB label 81312.bf)

• Label: 81312bv
• Number of curves: $4$
• Conductor: $81312$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 81312bv have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(7\)	\(1 - T\)
\(11\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 3 T + 5 T^{2}\)	1.5.ad
\(13\)	\( 1 + 13 T^{2}\)	1.13.a
\(17\)	\( 1 + 5 T + 17 T^{2}\)	1.17.f
\(19\)	\( 1 + 2 T + 19 T^{2}\)	1.19.c
\(23\)	\( 1 - 8 T + 23 T^{2}\)	1.23.ai
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 81312bv do not have complex multiplication.

Modular form 81312.2.a.bv

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 81312bv

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
81312.bf3	81312bv1	\([0, 1, 0, -391354, -94330744]\)	\(58465284603328/23532201\)	\(2668078690288704\)	\([2, 2]\)	\(614400\)	\(1.9234\)	\(\Gamma_0(N)\)-optimal
81312.bf4	81312bv2	\([0, 1, 0, -332064, -123833448]\)	\(-4464412682696/4706920449\)	\(-4269361509146055168\)	\([2]\)	\(1228800\)	\(2.2700\)
81312.bf2	81312bv3	\([0, 1, 0, -451249, -63604609]\)	\(1400416996672/570715299\)	\(4141289331964882944\)	\([4]\)	\(1228800\)	\(2.2700\)
81312.bf1	81312bv4	\([0, 1, 0, -6261064, -6032129380]\)	\(29925549856274696/4851\)	\(4400047314432\)	\([2]\)	\(1228800\)	\(2.2700\)