Elliptic curve isogeny class with Cremona label 227136cp (LMFDB label 227136.ba)

• Label: 227136cp
• Number of curves: $4$
• Conductor: $227136$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 227136cp have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(11\)	\( 1 + 2 T + 11 T^{2}\)	1.11.c
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 8 T + 23 T^{2}\)	1.23.i
\(29\)	\( 1 + 2 T + 29 T^{2}\)	1.29.c
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 227136cp do not have complex multiplication.

Modular form 227136.2.a.cp

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 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, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 227136cp

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
227136.ba3	227136cp1	\([0, -1, 0, -2084, 31290]\)	\(3241792/567\)	\(175155244992\)	\([2]\)	\(245760\)	\(0.87727\)	\(\Gamma_0(N)\)-optimal
227136.ba2	227136cp2	\([0, -1, 0, -9689, -335271]\)	\(5088448/441\)	\(8718838861824\)	\([2, 2]\)	\(491520\)	\(1.2238\)
227136.ba4	227136cp3	\([0, -1, 0, 10591, -1572351]\)	\(830584/7203\)	\(-1139261611278336\)	\([2]\)	\(983040\)	\(1.5704\)
227136.ba1	227136cp4	\([0, -1, 0, -151649, -22679775]\)	\(2438569736/21\)	\(3321462423552\)	\([2]\)	\(983040\)	\(1.5704\)