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

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

Rank

The elliptic curves in class 28224.bc 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
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 + 2 T + 13 T^{2}\)	1.13.c
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 - 4 T + 19 T^{2}\)	1.19.ae
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 - 2 T + 29 T^{2}\)	1.29.ac
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

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

Modular form 28224.2.a.bc

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 LMFDB 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 LMFDB labels.

Elliptic curves in class 28224.bc

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
28224.bc1	28224cl4	\([0, 0, 0, -57036, -5241040]\)	\(7301384/3\)	\(8431152758784\)	\([2]\)	\(73728\)	\(1.4426\)
28224.bc2	28224cl2	\([0, 0, 0, -4116, -54880]\)	\(21952/9\)	\(3161682284544\)	\([2, 2]\)	\(36864\)	\(1.0960\)
28224.bc3	28224cl1	\([0, 0, 0, -1911, 31556]\)	\(140608/3\)	\(16467095232\)	\([2]\)	\(18432\)	\(0.74947\)	\(\Gamma_0(N)\)-optimal
28224.bc4	28224cl3	\([0, 0, 0, 13524, -400624]\)	\(97336/81\)	\(-227641124487168\)	\([2]\)	\(73728\)	\(1.4426\)