Elliptic curve isogeny class with LMFDB label 80586.bc (Cremona label 80586bg)

• Label: 80586.bc
• Number of curves: $4$
• Conductor: $80586$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 80586.bc have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1\)
\(11\)	\(1\)
\(37\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(13\)	\( 1 - 4 T + 13 T^{2}\)	1.13.ae
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 + 2 T + 19 T^{2}\)	1.19.c
\(23\)	\( 1 - 6 T + 23 T^{2}\)	1.23.ag
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 80586.bc do not have complex multiplication.

Modular form 80586.2.a.bc

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 80586.bc

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
80586.bc1	80586bg4	\([1, -1, 1, -1176771245, -15537386236831]\)	\(139545621883503188502625/220644468\)	\(284955262959045492\)	\([2]\)	\(19906560\)	\(3.5060\)
80586.bc2	80586bg3	\([1, -1, 1, -73548905, -242753004007]\)	\(34069730739753390625/1354703543952\)	\(1749556234504791673488\)	\([2]\)	\(9953280\)	\(3.1594\)
80586.bc3	80586bg2	\([1, -1, 1, -14568665, -21184854919]\)	\(264788619837198625/3058196150592\)	\(3949562371408668387648\)	\([2]\)	\(6635520\)	\(2.9567\)
80586.bc4	80586bg1	\([1, -1, 1, -1674905, 306464249]\)	\(402355893390625/201513996288\)	\(260248871511136899072\)	\([2]\)	\(3317760\)	\(2.6101\)	\(\Gamma_0(N)\)-optimal