Elliptic curve isogeny class with Cremona label 31680cs (LMFDB label 31680.be)

• Label: 31680cs
• Number of curves: $2$
• Conductor: $31680$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 31680cs 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 \(\mathbb{F}_p\)
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 31680cs do not have complex multiplication.

Modular form 31680.2.a.cs

Isogeny matrix

The \((i,j)\)-th 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 31680cs

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
31680.be1	31680cs1	\([0, 0, 0, -588, 14992]\)	\(-117649/440\)	\(-84085309440\)	\([]\)	\(23040\)	\(0.78192\)	\(\Gamma_0(N)\)-optimal
31680.be2	31680cs2	\([0, 0, 0, 5172, -355952]\)	\(80062991/332750\)	\(-63589515264000\)	\([]\)	\(69120\)	\(1.3312\)