Elliptic curve isogeny class with Cremona label 31680dn (LMFDB label 31680.ea)

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

Rank

The elliptic curves in class 31680dn 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.ac
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(19\)	\( 1 + 4 T + 19 T^{2}\)	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 31680dn do not have complex multiplication.

Modular form 31680.2.a.dn

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 31680dn

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
31680.ea3	31680dn1	\([0, 0, 0, -12972, -479536]\)	\(1263214441/211200\)	\(40360948531200\)	\([2]\)	\(98304\)	\(1.3319\)	\(\Gamma_0(N)\)-optimal
31680.ea2	31680dn2	\([0, 0, 0, -59052, 5068496]\)	\(119168121961/10890000\)	\(2081111408640000\)	\([2, 2]\)	\(196608\)	\(1.6784\)
31680.ea4	31680dn3	\([0, 0, 0, 67668, 23873744]\)	\(179310732119/1392187500\)	\(-266051174400000000\)	\([2]\)	\(393216\)	\(2.0250\)
31680.ea1	31680dn4	\([0, 0, 0, -923052, 341337296]\)	\(455129268177961/4392300\)	\(839381601484800\)	\([2]\)	\(393216\)	\(2.0250\)