Elliptic curve isogeny class with Cremona label 56a (LMFDB label 56.a)

• Label: 56a
• Number of curves: $4$
• Conductor: $56$
• CM: no
• Rank: $0$

Rank

The elliptic curves in class 56a have rank \(0\).

L-function data

Bad L-factors:

Prime	L-Factor
\(2\)	\(1\)
\(7\)	\(1 + T\)

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 - 2 T + 5 T^{2}\)	1.5.ac
\(11\)	\( 1 + 4 T + 11 T^{2}\)	1.11.e
\(13\)	\( 1 - 2 T + 13 T^{2}\)	1.13.ac
\(17\)	\( 1 + 6 T + 17 T^{2}\)	1.17.g
\(19\)	\( 1 - 8 T + 19 T^{2}\)	1.19.ai
\(23\)	\( 1 + 23 T^{2}\)	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

The elliptic curves in class 56a do not have complex multiplication.

Modular form 56.2.a.a

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

Elliptic curves in class 56a

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality
56.a4	56a1	\([0, 0, 0, 1, 2]\)	\(432/7\)	\(-1792\)	\([4]\)	\(2\)	\(-0.69518\)	\(\Gamma_0(N)\)-optimal
56.a3	56a2	\([0, 0, 0, -19, 30]\)	\(740772/49\)	\(50176\)	\([2, 2]\)	\(4\)	\(-0.34861\)
56.a2	56a3	\([0, 0, 0, -59, -138]\)	\(11090466/2401\)	\(4917248\)	\([2]\)	\(8\)	\(-0.0020328\)
56.a1	56a4	\([0, 0, 0, -299, 1990]\)	\(1443468546/7\)	\(14336\)	\([2]\)	\(8\)	\(-0.0020328\)