Elliptic curve isogeny class with Cremona label 330096j (LMFDB label 330096.j)

• Label: 330096j
• Number of curves: $4$
• Conductor: $330096$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 330096j have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 + T\)
\(13\)	\(1 - T\)
\(23\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 330096j do not have complex multiplication.

Modular form 330096.2.a.j

Isogeny matrix

The \((i,j)\)-th entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona 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 Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 330096j

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
330096.j3	330096j1	\([0, -1, 0, -3879, 54918]\)	\(2725888/1053\)	\(2494108657872\)	\([2]\)	\(405504\)	\(1.0777\)	\(\Gamma_0(N)\)-optimal
330096.j2	330096j2	\([0, -1, 0, -27684, -1725696]\)	\(61918288/1521\)	\(57641622315264\)	\([2, 2]\)	\(811008\)	\(1.4243\)
330096.j4	330096j3	\([0, -1, 0, 4056, -5483712]\)	\(48668/85683\)	\(-12988578895039488\)	\([2]\)	\(1622016\)	\(1.7709\)
330096.j1	330096j4	\([0, -1, 0, -440304, -112307856]\)	\(62275269892/39\)	\(5911961263104\)	\([2]\)	\(1622016\)	\(1.7709\)