Elliptic curve isogeny class with Cremona label 206400hc (LMFDB label 206400.ic)

• Label: 206400hc
• Number of curves: $4$
• Conductor: $206400$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 206400hc have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1 - T\)
\(5\)	\(1\)
\(43\)	\(1 - T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(7\)	\( 1 + 2 T + 7 T^{2}\)	1.7.c
\(11\)	\( 1 - 6 T + 11 T^{2}\)	1.11.ag
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 2 T + 19 T^{2}\)	1.19.c
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 6 T + 29 T^{2}\)	1.29.g
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 206400hc do not have complex multiplication.

Modular form 206400.2.a.hc

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 206400hc

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
206400.ic3	206400hc1	\([0, 1, 0, -39233, -3002337]\)	\(1630532233/1161\)	\(4755456000000\)	\([2]\)	\(491520\)	\(1.3681\)	\(\Gamma_0(N)\)-optimal
206400.ic2	206400hc2	\([0, 1, 0, -47233, -1698337]\)	\(2845178713/1347921\)	\(5521084416000000\)	\([2, 2]\)	\(983040\)	\(1.7147\)
206400.ic1	206400hc3	\([0, 1, 0, -391233, 92901663]\)	\(1616855892553/22851963\)	\(93601640448000000\)	\([2]\)	\(1966080\)	\(2.0613\)
206400.ic4	206400hc4	\([0, 1, 0, 168767, -12714337]\)	\(129784785047/92307627\)	\(-378092040192000000\)	\([2]\)	\(1966080\)	\(2.0613\)