Elliptic curve isogeny class with Cremona label 23232cf (LMFDB label 23232.cp)

• Label: 23232cf
• Number of curves: $4$
• Conductor: $23232$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 23232cf have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(11\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + 3 T + 5 T^{2}\)	1.5.d
\(7\)	\( 1 - 2 T + 7 T^{2}\)	1.7.ac
\(13\)	\( 1 - 5 T + 13 T^{2}\)	1.13.af
\(17\)	\( 1 - 3 T + 17 T^{2}\)	1.17.ad
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 9 T + 29 T^{2}\)	1.29.j
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 23232cf do not have complex multiplication.

Modular form 23232.2.a.cf

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 23232cf

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
23232.cp4	23232cf1	\([0, 1, 0, -629724, 152513370]\)	\(243578556889408/52089208083\)	\(5905869411886564032\)	\([2]\)	\(460800\)	\(2.3162\)	\(\Gamma_0(N)\)-optimal
23232.cp2	23232cf2	\([0, 1, 0, -9487529, 11244256791]\)	\(13015685560572352/864536409\)	\(6273347523643183104\)	\([2, 2]\)	\(921600\)	\(2.6627\)
23232.cp3	23232cf3	\([0, 1, 0, -8901889, 12693481535]\)	\(-1343891598641864/421900912521\)	\(-24491563499081409527808\)	\([2]\)	\(1843200\)	\(3.0093\)
23232.cp1	23232cf4	\([0, 1, 0, -151798049, 719808335871]\)	\(6663712298552914184/29403\)	\(1706859170463744\)	\([2]\)	\(1843200\)	\(3.0093\)