Elliptic curve isogeny class with LMFDB label 336336.gn (Cremona label 336336gn)

• Label: 336336.gn
• Number of curves: $4$
• Conductor: $336336$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 336336.gn have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(7\)	\(1\)
\(11\)	\(1 - T\)
\(13\)	\(1 + T\)

Good L-factors:

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

See L-function page for more information

Complex multiplication

The elliptic curves in class 336336.gn do not have complex multiplication.

Modular form 336336.2.a.gn

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 336336.gn

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
336336.gn1	336336gn4	\([0, 1, 0, -51097608, -140590669836]\)	\(30618029936661765625/3678951124992\)	\(1772850876023536877568\)	\([2]\)	\(23887872\)	\(3.1019\)
336336.gn2	336336gn3	\([0, 1, 0, -2928648, -2576965644]\)	\(-5764706497797625/2612665516032\)	\(-1259018179770977353728\)	\([2]\)	\(11943936\)	\(2.7554\)
336336.gn3	336336gn2	\([0, 1, 0, -1411608, 367319700]\)	\(645532578015625/252306960048\)	\(121584277678846574592\)	\([2]\)	\(7962624\)	\(2.5526\)
336336.gn4	336336gn1	\([0, 1, 0, 281832, 41501844]\)	\(5137417856375/4510142208\)	\(-2173393799696351232\)	\([2]\)	\(3981312\)	\(2.2060\)	\(\Gamma_0(N)\)-optimal