Elliptic curve isogeny class with LMFDB label 1680.f (Cremona label 1680n)

• Label: 1680.f
• Number of curves: $4$
• Conductor: $1680$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 1680.f have rank \(1\).

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 \(\mathbb{F}_p\)
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 - 8 T + 19 T^{2}\)	1.19.ai
\(23\)	\( 1 + 8 T + 23 T^{2}\)	1.23.i
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 1680.2.a.f

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 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, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 1680.f

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
1680.f1	1680n3	\([0, -1, 0, -1800, 30000]\)	\(157551496201/13125\)	\(53760000\)	\([4]\)	\(1024\)	\(0.52719\)
1680.f2	1680n2	\([0, -1, 0, -120, 432]\)	\(47045881/11025\)	\(45158400\)	\([2, 2]\)	\(512\)	\(0.18061\)
1680.f3	1680n1	\([0, -1, 0, -40, -80]\)	\(1771561/105\)	\(430080\)	\([2]\)	\(256\)	\(-0.16596\)	\(\Gamma_0(N)\)-optimal
1680.f4	1680n4	\([0, -1, 0, 280, 2352]\)	\(590589719/972405\)	\(-3982970880\)	\([2]\)	\(1024\)	\(0.52719\)