Elliptic curve isogeny class with LMFDB label 1216.e (Cremona label 1216b)

• Label: 1216.e
• Number of curves: $3$
• Conductor: $1216$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 1216.e have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(19\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + T + 3 T^{2}\)	1.3.b
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(7\)	\( 1 + T + 7 T^{2}\)	1.7.b
\(11\)	\( 1 - 6 T + 11 T^{2}\)	1.11.ag
\(13\)	\( 1 + 5 T + 13 T^{2}\)	1.13.f
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(23\)	\( 1 - 3 T + 23 T^{2}\)	1.23.ad
\(29\)	\( 1 + 9 T + 29 T^{2}\)	1.29.j
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 1216.e do not have complex multiplication.

Modular form 1216.2.a.e

Isogeny matrix

The \((i,j)\)-th 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.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 1216.e

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
1216.e1	1216b3	\([0, -1, 0, -5473, -1251871]\)	\(-69173457625/2550136832\)	\(-668503069687808\)	\([]\)	\(3456\)	\(1.5249\)
1216.e2	1216b1	\([0, -1, 0, -993, 12385]\)	\(-413493625/152\)	\(-39845888\)	\([]\)	\(384\)	\(0.42628\)	\(\Gamma_0(N)\)-optimal
1216.e3	1216b2	\([0, -1, 0, 607, 45601]\)	\(94196375/3511808\)	\(-920599396352\)	\([]\)	\(1152\)	\(0.97558\)