Elliptic curve isogeny class with LMFDB label 8400.cf (Cremona label 8400v)

• Label: 8400.cf
• Number of curves: $4$
• Conductor: $8400$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 8400.cf have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(5\)	\(1\)
\(7\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(11\)	\( 1 - 4 T + 11 T^{2}\)	1.11.ae
\(13\)	\( 1 - 6 T + 13 T^{2}\)	1.13.ag
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 - 8 T + 19 T^{2}\)	1.19.ai
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 8400.cf do not have complex multiplication.

Modular form 8400.2.a.cf

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 8400.cf

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
8400.cf1	8400v3	\([0, 1, 0, -72408, -7522812]\)	\(2624033547076/324135\)	\(5186160000000\)	\([2]\)	\(36864\)	\(1.4630\)
8400.cf2	8400v2	\([0, 1, 0, -4908, -97812]\)	\(3269383504/893025\)	\(3572100000000\)	\([2, 2]\)	\(18432\)	\(1.1164\)
8400.cf3	8400v1	\([0, 1, 0, -1783, 27188]\)	\(2508888064/118125\)	\(29531250000\)	\([2]\)	\(9216\)	\(0.76986\)	\(\Gamma_0(N)\)-optimal
8400.cf4	8400v4	\([0, 1, 0, 12592, -622812]\)	\(13799183324/18600435\)	\(-297606960000000\)	\([4]\)	\(36864\)	\(1.4630\)