Elliptic curve isogeny class with LMFDB label 317680.bl (Cremona label 317680bl)

• Label: 317680.bl
• Number of curves: $4$
• Conductor: $317680$
• CM: no
• Rank: $2$

Rank

The elliptic curves in class 317680.bl have rank \(2\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(5\)	\(1 - T\)
\(11\)	\(1 - T\)
\(19\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 317680.bl do not have complex multiplication.

Modular form 317680.2.a.bl

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 317680.bl

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
317680.bl1	317680bl4	\([0, 0, 0, -2724467, -89070974]\)	\(46424454082884/26794860125\)	\(1290841908072852608000\)	\([2]\)	\(15925248\)	\(2.7403\)
317680.bl2	317680bl2	\([0, 0, 0, -1821967, 943208526]\)	\(55537159171536/228765625\)	\(2755194974884000000\)	\([2, 2]\)	\(7962624\)	\(2.3938\)
317680.bl3	317680bl1	\([0, 0, 0, -1820162, 945177059]\)	\(885956203616256/15125\)	\(11385103202000\)	\([2]\)	\(3981312\)	\(2.0472\)	\(\Gamma_0(N)\)-optimal
317680.bl4	317680bl3	\([0, 0, 0, -948347, 1849501914]\)	\(-1957960715364/29541015625\)	\(-1423137900250000000000\)	\([2]\)	\(15925248\)	\(2.7403\)