Elliptic curve isogeny class with LMFDB label 256.b (Cremona label 256b)

• Label: 256.b
• Number of curves: $2$
• Conductor: $256$
• CM: \(\Q(\sqrt{-1}) \)
• Rank: $1$

Rank

The elliptic curves in class 256.b have rank \(1\).

L-function data

Bad L-factors:

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

Good L-factors:

Prime	L-Factor	Isogeny Class over \(\mathbb{F}_p\)
\(3\)	\( 1 + 3 T^{2}\)	1.3.a
\(5\)	\( 1 + 4 T + 5 T^{2}\)	1.5.e
\(7\)	\( 1 + 7 T^{2}\)	1.7.a
\(11\)	\( 1 + 11 T^{2}\)	1.11.a
\(13\)	\( 1 + 4 T + 13 T^{2}\)	1.13.e
\(17\)	\( 1 + 2 T + 17 T^{2}\)	1.17.c
\(19\)	\( 1 + 19 T^{2}\)	1.19.a
\(23\)	\( 1 + 23 T^{2}\)	1.23.a
\(29\)	\( 1 + 4 T + 29 T^{2}\)	1.29.e
$\cdots$	$\cdots$	$\cdots$

See L-function page for more information

Complex multiplication

Each elliptic curve in class 256.b has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-1}) \).

Modular form 256.2.a.b

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.

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

Isogeny graph

The vertices are labelled with LMFDB labels.

Elliptic curves in class 256.b

LMFDB label	Cremona label	Weierstrass coefficients	j-invariant	Discriminant	Torsion structure	Modular degree	Faltings height	Optimality	CM discriminant
256.b1	256b1	\([0, 0, 0, -2, 0]\)	\(1728\)	\(512\)	\([2]\)	\(8\)	\(-0.79067\)	\(\Gamma_0(N)\)-optimal	\(-4\)
256.b2	256b2	\([0, 0, 0, 8, 0]\)	\(1728\)	\(-32768\)	\([2]\)	\(16\)	\(-0.44410\)		\(-4\)