Elliptic curve isogeny class with LMFDB label 69696.fg (Cremona label 69696cn)

• Label: 69696.fg
• Number of curves: $4$
• Conductor: $69696$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 69696.fg have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 69696.fg do not have complex multiplication.

Modular form 69696.2.a.fg

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 69696.fg

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
69696.fg1	69696cn4	\([0, 0, 0, -1366182444, 19436191250960]\)	\(6663712298552914184/29403\)	\(1244300335268069376\)	\([2]\)	\(14745600\)	\(3.5586\)
69696.fg2	69696cn2	\([0, 0, 0, -85387764, 303680321120]\)	\(13015685560572352/864536409\)	\(4573270344735880482816\)	\([2, 2]\)	\(7372800\)	\(3.2120\)
69696.fg3	69696cn3	\([0, 0, 0, -80117004, 342804118448]\)	\(-1343891598641864/421900912521\)	\(-17854349790830347545772032\)	\([2]\)	\(14745600\)	\(3.5586\)
69696.fg4	69696cn1	\([0, 0, 0, -5667519, 4123528508]\)	\(243578556889408/52089208083\)	\(4305378801265305179328\)	\([2]\)	\(3686400\)	\(2.8655\)	\(\Gamma_0(N)\)-optimal