Elliptic curve isogeny class with LMFDB label 85680.dp (Cremona label 85680fi)

• Label: 85680.dp
• Number of curves: $4$
• Conductor: $85680$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 85680.dp have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(5\)	\(1 - T\)
\(7\)	\(1 + T\)
\(17\)	\(1 - T\)

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
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 85680.dp do not have complex multiplication.

Modular form 85680.2.a.dp

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 85680.dp

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
85680.dp1	85680fi4	\([0, 0, 0, -242787, 46043426]\)	\(530044731605089/26309115\)	\(78558596444160\)	\([2]\)	\(524288\)	\(1.7376\)
85680.dp2	85680fi3	\([0, 0, 0, -77187, -7672894]\)	\(17032120495489/1339001685\)	\(3998237607383040\)	\([2]\)	\(524288\)	\(1.7376\)
85680.dp3	85680fi2	\([0, 0, 0, -15987, 638066]\)	\(151334226289/28676025\)	\(85626151833600\)	\([2, 2]\)	\(262144\)	\(1.3910\)
85680.dp4	85680fi1	\([0, 0, 0, 2013, 58466]\)	\(302111711/669375\)	\(-1998743040000\)	\([2]\)	\(131072\)	\(1.0445\)	\(\Gamma_0(N)\)-optimal