Elliptic curve isogeny class with Cremona label 67032cf (LMFDB label 67032.ck)

• Label: 67032cf
• Number of curves: $2$
• Conductor: $67032$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 67032cf have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(7\)	\(1\)
\(19\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(11\)	\( 1 - T + 11 T^{2}\)	1.11.ab
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 3 T + 17 T^{2}\)	1.17.d
\(23\)	\( 1 + 3 T + 23 T^{2}\)	1.23.d
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 67032cf do not have complex multiplication.

Modular form 67032.2.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.

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

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 67032cf

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
67032.ck1	67032cf1	\([0, 0, 0, -2185328019, -39309909712738]\)	\(13141891860831409148932/4237307541832617\)	\(372139449699357482361504768\)	\([2]\)	\(36126720\)	\(4.0728\)	\(\Gamma_0(N)\)-optimal
67032.ck2	67032cf2	\([0, 0, 0, -1888852539, -50359372967050]\)	\(-4242991426585187031506/3781894171664380023\)	\(-664285989142859752330446403584\)	\([2]\)	\(72253440\)	\(4.4194\)