Elliptic curve isogeny class with LMFDB label 31680.cj (Cremona label 31680ca)

• Label: 31680.cj
• Number of curves: $4$
• Conductor: $31680$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 31680.cj have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(5\)	\(1 - T\)
\(11\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(13\)	\( 1 - 6 T + 13 T^{2}\)	1.13.ag
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 - 8 T + 23 T^{2}\)	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 31680.cj do not have complex multiplication.

Modular form 31680.2.a.cj

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 31680.cj

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
31680.cj1	31680ca4	\([0, 0, 0, -1134732, 458163056]\)	\(3382175663521924/59189241375\)	\(2827810043486208000\)	\([2]\)	\(786432\)	\(2.3371\)
31680.cj2	31680ca2	\([0, 0, 0, -144732, -10304944]\)	\(28071778927696/12404390625\)	\(148157247744000000\)	\([2, 2]\)	\(393216\)	\(1.9905\)
31680.cj3	31680ca1	\([0, 0, 0, -122952, -16586296]\)	\(275361373935616/148240125\)	\(110660660352000\)	\([2]\)	\(196608\)	\(1.6439\)	\(\Gamma_0(N)\)-optimal
31680.cj4	31680ca3	\([0, 0, 0, 496788, -76766416]\)	\(283811208976796/217529296875\)	\(-10392624000000000000\)	\([2]\)	\(786432\)	\(2.3371\)