Elliptic curve isogeny class with Cremona label 121680cx (LMFDB label 121680.ex)

• Label: 121680cx
• Number of curves: $4$
• Conductor: $121680$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 121680cx have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 + 5 T + 7 T^{2}\)	1.7.f
\(11\)	\( 1 + T + 11 T^{2}\)	1.11.b
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(19\)	\( 1 - 6 T + 19 T^{2}\)	1.19.ag
\(23\)	\( 1 + 7 T + 23 T^{2}\)	1.23.h
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 121680cx do not have complex multiplication.

Modular form 121680.2.a.cx

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 Cremona 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 Cremona labels.

Elliptic curves in class 121680cx

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
121680.ex3	121680cx1	\([0, 0, 0, -20787, -1313806]\)	\(-1860867/320\)	\(-170818067496960\)	\([2]\)	\(414720\)	\(1.4573\)	\(\Gamma_0(N)\)-optimal
121680.ex2	121680cx2	\([0, 0, 0, -345267, -78085774]\)	\(8527173507/200\)	\(106761292185600\)	\([2]\)	\(829440\)	\(1.8038\)
121680.ex4	121680cx3	\([0, 0, 0, 141453, 5575986]\)	\(804357/500\)	\(-194572455008256000\)	\([2]\)	\(1244160\)	\(2.0066\)
121680.ex1	121680cx4	\([0, 0, 0, -588627, 45438354]\)	\(57960603/31250\)	\(12160778438016000000\)	\([2]\)	\(2488320\)	\(2.3531\)