Elliptic curve isogeny class with Cremona label 81312bn (LMFDB label 81312.be)

• Label: 81312bn
• Number of curves: $4$
• Conductor: $81312$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 81312bn have rank \(0\).

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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(19\)	\( 1 - 6 T + 19 T^{2}\)	1.19.ag
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 81312.2.a.bn

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 81312bn

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
81312.be3	81312bn1	\([0, 1, 0, -6574, 165416]\)	\(277167808/53361\)	\(6050065057344\)	\([2, 2]\)	\(153600\)	\(1.1698\)	\(\Gamma_0(N)\)-optimal
81312.be4	81312bn2	\([0, 1, 0, 13391, 991967]\)	\(36594368/79233\)	\(-574939515752448\)	\([2]\)	\(307200\)	\(1.5163\)
81312.be2	81312bn3	\([0, 1, 0, -31984, -2060500]\)	\(3989418056/307461\)	\(278879189309952\)	\([2]\)	\(307200\)	\(1.5163\)
81312.be1	81312bn4	\([0, 1, 0, -99744, 12091176]\)	\(120993582536/6237\)	\(5657203689984\)	\([4]\)	\(307200\)	\(1.5163\)