Elliptic curve isogeny class with LMFDB label 6720.bc (Cremona label 6720k)

• Label: 6720.bc
• Number of curves: $4$
• Conductor: $6720$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 6720.bc 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
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 6720.bc do not have complex multiplication.

Modular form 6720.2.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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 6720.bc

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
6720.bc1	6720k3	\([0, -1, 0, -2625, 52545]\)	\(15267472418/36015\)	\(4720558080\)	\([4]\)	\(4096\)	\(0.73620\)
6720.bc2	6720k2	\([0, -1, 0, -225, 225]\)	\(19307236/11025\)	\(722534400\)	\([2, 2]\)	\(2048\)	\(0.38963\)
6720.bc3	6720k1	\([0, -1, 0, -145, -623]\)	\(20720464/105\)	\(1720320\)	\([2]\)	\(1024\)	\(0.043052\)	\(\Gamma_0(N)\)-optimal
6720.bc4	6720k4	\([0, -1, 0, 895, 897]\)	\(604223422/354375\)	\(-46448640000\)	\([2]\)	\(4096\)	\(0.73620\)