Elliptic curve isogeny class with Cremona label 28224gd (LMFDB label 28224.by)

• Label: 28224gd
• Number of curves: $4$
• Conductor: $28224$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 28224gd have rank \(0\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(5\)	\( 1 - 4 T + 5 T^{2}\)	1.5.ae
\(11\)	\( 1 + 6 T + 11 T^{2}\)	1.11.g
\(13\)	\( 1 - 5 T + 13 T^{2}\)	1.13.af
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + T + 19 T^{2}\)	1.19.b
\(23\)	\( 1 + 6 T + 23 T^{2}\)	1.23.g
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 28224.2.a.gd

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 28224gd

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
28224.by4	28224gd1	\([0, 0, 0, 1764, -148176]\)	\(432/7\)	\(-9836344885248\)	\([2]\)	\(49152\)	\(1.1737\)	\(\Gamma_0(N)\)-optimal
28224.by3	28224gd2	\([0, 0, 0, -33516, -2222640]\)	\(740772/49\)	\(275417656786944\)	\([2, 2]\)	\(98304\)	\(1.5202\)
28224.by2	28224gd3	\([0, 0, 0, -104076, 10224144]\)	\(11090466/2401\)	\(26990930365120512\)	\([2]\)	\(196608\)	\(1.8668\)
28224.by1	28224gd4	\([0, 0, 0, -527436, -147435120]\)	\(1443468546/7\)	\(78690759081984\)	\([2]\)	\(196608\)	\(1.8668\)