Elliptic curve isogeny class with Cremona label 28224ck (LMFDB label 28224.bd)

• Label: 28224ck
• Number of curves: $4$
• Conductor: $28224$
• CM: no
• Rank: $1$

Rank

The elliptic curves in class 28224ck have rank \(1\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(3\)	\(1\)
\(7\)	\(1\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 + T + 5 T^{2}\)	1.5.b
\(11\)	\( 1 - T + 11 T^{2}\)	1.11.ab
\(13\)	\( 1 + 13 T^{2}\)	1.13.a
\(17\)	\( 1 + 8 T + 17 T^{2}\)	1.17.i
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 4 T + 23 T^{2}\)	1.23.e
\(29\)	\( 1 + 5 T + 29 T^{2}\)	1.29.f
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

The elliptic curves in class 28224ck do not have complex multiplication.

Modular form 28224.2.a.ck

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 28224ck

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
28224.bd4	28224ck1	\([0, 0, 0, 1764, 148176]\)	\(432/7\)	\(-9836344885248\)	\([2]\)	\(49152\)	\(1.1737\)	\(\Gamma_0(N)\)-optimal
28224.bd3	28224ck2	\([0, 0, 0, -33516, 2222640]\)	\(740772/49\)	\(275417656786944\)	\([2, 2]\)	\(98304\)	\(1.5202\)
28224.bd2	28224ck3	\([0, 0, 0, -104076, -10224144]\)	\(11090466/2401\)	\(26990930365120512\)	\([2]\)	\(196608\)	\(1.8668\)
28224.bd1	28224ck4	\([0, 0, 0, -527436, 147435120]\)	\(1443468546/7\)	\(78690759081984\)	\([2]\)	\(196608\)	\(1.8668\)