Elliptic curve isogeny class with LMFDB label 8712.z (Cremona label 8712g)

• Label: 8712.z
• Number of curves: $2$
• Conductor: $8712$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 8712.z have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(11\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 4 T + 5 T^{2}\)	1.5.ae
\(7\)	\( 1 - 4 T + 7 T^{2}\)	1.7.ae
\(13\)	\( 1 + 6 T + 13 T^{2}\)	1.13.g
\(17\)	\( 1 - 6 T + 17 T^{2}\)	1.17.ag
\(19\)	\( 1 - 6 T + 19 T^{2}\)	1.19.ag
\(23\)	\( 1 + 2 T + 23 T^{2}\)	1.23.c
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 8712.z do not have complex multiplication.

Modular form 8712.2.a.z

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 LMFDB numbering.

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

Isogeny graph

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

Elliptic curves in class 8712.z

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
8712.z1	8712g2	\([0, 0, 0, -12243, -15730]\)	\(102129622/59049\)	\(117340540213248\)	\([2]\)	\(46080\)	\(1.3892\)
8712.z2	8712g1	\([0, 0, 0, -8283, 289190]\)	\(63253004/243\)	\(241441440768\)	\([2]\)	\(23040\)	\(1.0426\)	\(\Gamma_0(N)\)-optimal