Elliptic curve isogeny class with LMFDB label 53312.bn (Cremona label 53312bs)

• Label: 53312.bn
• Number of curves: $4$
• Conductor: $53312$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 53312.bn have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 53312.bn do not have complex multiplication.

Modular form 53312.2.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 LMFDB numbering.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 53312.bn

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
53312.bn1	53312bs4	\([0, 0, 0, -1657964, -821339568]\)	\(16342588257633/8185058\)	\(252435205624168448\)	\([2]\)	\(589824\)	\(2.2918\)
53312.bn2	53312bs2	\([0, 0, 0, -121324, -8149680]\)	\(6403769793/2775556\)	\(85600865574977536\)	\([2, 2]\)	\(294912\)	\(1.9452\)
53312.bn3	53312bs1	\([0, 0, 0, -58604, 5372752]\)	\(721734273/13328\)	\(411048574189568\)	\([2]\)	\(147456\)	\(1.5987\)	\(\Gamma_0(N)\)-optimal
53312.bn4	53312bs3	\([0, 0, 0, 411796, -60395440]\)	\(250404380127/196003234\)	\(-6044931713103560704\)	\([2]\)	\(589824\)	\(2.2918\)