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

• Label: 28224di
• Number of curves: $4$
• Conductor: $28224$
• CM: \(\Q(\sqrt{-3}) \)
• Rank: $1$

Rank

The elliptic curves in class 28224di 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 \(\mathbb{F}_p\)
\(5\)	\( 1 + 5 T^{2}\)	1.5.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 17 T^{2}\)	1.17.a
\(19\)	\( 1 + 8 T + 19 T^{2}\)	1.19.i
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 29 T^{2}\)	1.29.a
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 28224di has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).

Modular form 28224.2.a.di

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

The vertices are labelled with Cremona labels.

Elliptic curves in class 28224di

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
28224.di4	28224di1	\([0, 0, 0, 0, 2744]\)	\(0\)	\(-3252759552\)	\([2]\)	\(9216\)	\(0.50411\)	\(\Gamma_0(N)\)-optimal	\(-3\)
28224.di2	28224di2	\([0, 0, 0, -2940, 60368]\)	\(54000\)	\(52044152832\)	\([2]\)	\(18432\)	\(0.85069\)		\(-12\)
28224.di3	28224di3	\([0, 0, 0, 0, -74088]\)	\(0\)	\(-2371261713408\)	\([2]\)	\(27648\)	\(1.0534\)		\(-3\)
28224.di1	28224di4	\([0, 0, 0, -26460, -1629936]\)	\(54000\)	\(37940187414528\)	\([2]\)	\(55296\)	\(1.4000\)		\(-12\)