Elliptic curve isogeny class with Cremona label 1680l (LMFDB label 1680.a)

• Label: 1680l
• Number of curves: $4$
• Conductor: $1680$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 1680l have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(5\)	\(1 + T\)
\(7\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(11\)	\( 1 - 4 T + 11 T^{2}\)	1.11.ae
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 1680l do not have complex multiplication.

Modular form 1680.2.a.l

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 1680l

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
1680.a2	1680l1	\([0, -1, 0, -61, -164]\)	\(1594753024/4725\)	\(75600\)	\([2]\)	\(288\)	\(-0.19410\)	\(\Gamma_0(N)\)-optimal
1680.a3	1680l2	\([0, -1, 0, -36, -324]\)	\(-20720464/178605\)	\(-45722880\)	\([2]\)	\(576\)	\(0.15247\)
1680.a1	1680l3	\([0, -1, 0, -301, 1960]\)	\(189123395584/16078125\)	\(257250000\)	\([2]\)	\(864\)	\(0.35520\)
1680.a4	1680l4	\([0, -1, 0, 324, 8460]\)	\(14647977776/132355125\)	\(-33882912000\)	\([2]\)	\(1728\)	\(0.70178\)