Elliptic curve isogeny class with LMFDB label 227136.fx (Cremona label 227136r)

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

Rank

The elliptic curves in class 227136.fx 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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(11\)	\( 1 - 4 T + 11 T^{2}\)	1.11.ae
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 - 4 T + 23 T^{2}\)	1.23.ae
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 227136.2.a.fx

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 227136.fx

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
227136.fx1	227136r4	\([0, 1, 0, -21161729, -37468729089]\)	\(828279937799497/193444524\)	\(244769035241105915904\)	\([2]\)	\(12386304\)	\(2.9028\)
227136.fx2	227136r2	\([0, 1, 0, -1476609, -441018369]\)	\(281397674377/96589584\)	\(122216637623816945664\)	\([2, 2]\)	\(6193152\)	\(2.5563\)
227136.fx3	227136r1	\([0, 1, 0, -611329, 178695167]\)	\(19968681097/628992\)	\(795875540081836032\)	\([2]\)	\(3096576\)	\(2.2097\)	\(\Gamma_0(N)\)-optimal
227136.fx4	227136r3	\([0, 1, 0, 4364031, -3061129473]\)	\(7264187703863/7406095788\)	\(-9371073853359519694848\)	\([2]\)	\(12386304\)	\(2.9028\)