Elliptic curve isogeny class with Cremona label 28224bn (LMFDB label 28224.dw)

• Label: 28224bn
• Number of curves: $4$
• Conductor: $28224$
• CM: \(\Q(\sqrt{-7}) \)
• Rank: $1$

Rank

The elliptic curves in class 28224bn 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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 2 T + 5 T^{2}\)	1.5.c
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 - 2 T + 17 T^{2}\)	1.17.ac
\(19\)	\( 1 + 4 T + 19 T^{2}\)	1.19.e
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 6 T + 29 T^{2}\)	1.29.ag
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 28224bn has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-7}) \).

Modular form 28224.2.a.bn

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 28224bn

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
28224.dw4	28224bn1	\([0, 0, 0, -1260, 21168]\)	\(-3375\)	\(-65548320768\)	\([2]\)	\(16384\)	\(0.78989\)	\(\Gamma_0(N)\)-optimal	\(-7\)
28224.dw3	28224bn2	\([0, 0, 0, -21420, 1206576]\)	\(16581375\)	\(65548320768\)	\([2]\)	\(32768\)	\(1.1365\)		\(-28\)
28224.dw2	28224bn3	\([0, 0, 0, -61740, -7260624]\)	\(-3375\)	\(-7711694390034432\)	\([2]\)	\(114688\)	\(1.7628\)		\(-7\)
28224.dw1	28224bn4	\([0, 0, 0, -1049580, -413855568]\)	\(16581375\)	\(7711694390034432\)	\([2]\)	\(229376\)	\(2.1094\)		\(-28\)