Elliptic curve isogeny class with Cremona label 8624r (LMFDB label 8624.j)

• Label: 8624r
• Number of curves: $3$
• Conductor: $8624$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 8624r have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(7\)	\(1\)
\(11\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 - 4 T + 5 T^{2}\)	1.5.ae
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 - 4 T + 17 T^{2}\)	1.17.ae
\(19\)	\( 1 + 6 T + 19 T^{2}\)	1.19.g
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 8624r do not have complex multiplication.

Modular form 8624.2.a.r

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.

\(\left(\begin{array}{rrr} 1 & 5 & 25 \\ 5 & 1 & 5 \\ 25 & 5 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 8624r

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
8624.j3	8624r1	\([0, -1, 0, -261, 3949]\)	\(-4096/11\)	\(-5300793344\)	\([]\)	\(2880\)	\(0.55337\)	\(\Gamma_0(N)\)-optimal
8624.j2	8624r2	\([0, -1, 0, -8101, -505651]\)	\(-122023936/161051\)	\(-77608915349504\)	\([]\)	\(14400\)	\(1.3581\)
8624.j1	8624r3	\([0, -1, 0, -6131141, -5841282131]\)	\(-52893159101157376/11\)	\(-5300793344\)	\([]\)	\(72000\)	\(2.1628\)