Elliptic curve isogeny class with LMFDB label 12870.ce (Cremona label 12870cf)

• Label: 12870.ce
• Number of curves: $4$
• Conductor: $12870$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 12870.ce have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1 - T\)
\(3\)	\(1\)
\(5\)	\(1 - T\)
\(11\)	\(1 - T\)
\(13\)	\(1 - T\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 12870.ce do not have complex multiplication.

Modular form 12870.2.a.ce

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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 12870.ce

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
12870.ce1	12870cf4	\([1, -1, 1, -343337, 77519211]\)	\(6139836723518159689/3799803150\)	\(2770056496350\)	\([2]\)	\(98304\)	\(1.7084\)
12870.ce2	12870cf3	\([1, -1, 1, -48317, -2338941]\)	\(17111482619973769/6627044531250\)	\(4831115463281250\)	\([2]\)	\(98304\)	\(1.7084\)
12870.ce3	12870cf2	\([1, -1, 1, -21587, 1200111]\)	\(1525998818291689/37268302500\)	\(27168592522500\)	\([2, 2]\)	\(49152\)	\(1.3618\)
12870.ce4	12870cf1	\([1, -1, 1, 193, 58839]\)	\(1095912791/2055596400\)	\(-1498529775600\)	\([4]\)	\(24576\)	\(1.0153\)	\(\Gamma_0(N)\)-optimal