Elliptic curve isogeny class with Cremona label 212160ei (LMFDB label 212160.gg)

• Label: 212160ei
• Number of curves: $4$
• Conductor: $212160$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 212160ei have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(5\)	\(1 + T\)
\(13\)	\(1 + T\)
\(17\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 + 2 T + 7 T^{2}\)	1.7.c
\(11\)	\( 1 + T + 11 T^{2}\)	1.11.b
\(19\)	\( 1 + 3 T + 19 T^{2}\)	1.19.d
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 212160ei do not have complex multiplication.

Modular form 212160.2.a.ei

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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 212160ei

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
212160.gg3	212160ei1	\([0, 1, 0, -1665, 24543]\)	\(1948441249/89505\)	\(23463198720\)	\([2]\)	\(229376\)	\(0.75153\)	\(\Gamma_0(N)\)-optimal
212160.gg2	212160ei2	\([0, 1, 0, -4545, -86625]\)	\(39616946929/10989225\)	\(2880759398400\)	\([2, 2]\)	\(458752\)	\(1.0981\)
212160.gg4	212160ei3	\([0, 1, 0, 11775, -553377]\)	\(688699320191/910381875\)	\(-238651146240000\)	\([2]\)	\(917504\)	\(1.4447\)
212160.gg1	212160ei4	\([0, 1, 0, -66945, -6688545]\)	\(126574061279329/16286595\)	\(4269433159680\)	\([2]\)	\(917504\)	\(1.4447\)