Elliptic curve isogeny class with LMFDB label 247744.cb (Cremona label 247744cb)

• Label: 247744.cb
• Number of curves: $3$
• Conductor: $247744$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 247744.cb have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(7\)	\(1\)
\(79\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 - T + 3 T^{2}\)	1.3.ab
\(5\)	\( 1 - 3 T + 5 T^{2}\)	1.5.ad
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 5 T + 13 T^{2}\)	1.13.af
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 - 2 T + 19 T^{2}\)	1.19.ac
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 247744.cb do not have complex multiplication.

Modular form 247744.2.a.cb

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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 247744.cb

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
247744.cb1	247744cb3	\([0, 1, 0, -16359009, 25461859583]\)	\(15698803397448457/20709376\)	\(638697439762579456\)	\([]\)	\(8709120\)	\(2.6925\)
247744.cb2	247744cb2	\([0, 1, 0, -255649, 14834623]\)	\(59914169497/31554496\)	\(973171562880434176\)	\([]\)	\(2903040\)	\(2.1432\)
247744.cb3	247744cb1	\([0, 1, 0, -145889, -21495937]\)	\(11134383337/316\)	\(9745749508096\)	\([]\)	\(967680\)	\(1.5939\)	\(\Gamma_0(N)\)-optimal