Elliptic curve isogeny class with Cremona label 186576cf (LMFDB label 186576.bw)

• Label: 186576cf
• Number of curves: $4$
• Conductor: $186576$
• CM: no
• Rank: $2$

Rank

The elliptic curves in class 186576cf have rank \(2\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(13\)	\(1\)
\(23\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + T + 5 T^{2}\)	1.5.b
\(7\)	\( 1 + 4 T + 7 T^{2}\)	1.7.e
\(11\)	\( 1 - 4 T + 11 T^{2}\)	1.11.ae
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 - 5 T + 19 T^{2}\)	1.19.af
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 186576cf do not have complex multiplication.

Modular form 186576.2.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 186576cf

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
186576.bw3	186576cf1	\([0, 1, 0, -35039, 2512836]\)	\(61604313088/621\)	\(47959174224\)	\([2]\)	\(442368\)	\(1.2084\)	\(\Gamma_0(N)\)-optimal
186576.bw2	186576cf2	\([0, 1, 0, -35884, 2384396]\)	\(4135597648/385641\)	\(476522355089664\)	\([2, 2]\)	\(884736\)	\(1.5550\)
186576.bw4	186576cf3	\([0, 1, 0, 41856, 11371140]\)	\(1640689628/12223143\)	\(-60414747280063488\)	\([2]\)	\(1769472\)	\(1.9015\)
186576.bw1	186576cf4	\([0, 1, 0, -127144, -14808988]\)	\(45989074372/7555707\)	\(37345233458138112\)	\([2]\)	\(1769472\)	\(1.9015\)