# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 256.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality CM discriminant
256.b1 256b1 [0, 0, 0, -2, 0] [2] 8 $$\Gamma_0(N)$$-optimal -4
256.b2 256b2 [0, 0, 0, 8, 0] [2] 16   -4

## Rank

sage: E.rank()

The elliptic curves in class 256.b have rank $$1$$.

## Complex multiplication

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

## Modular form256.2.a.b

sage: E.q_eigenform(10)

$$q - 4q^{5} - 3q^{9} - 4q^{13} - 2q^{17} + O(q^{20})$$

## Isogeny matrix

sage: E.isogeny_class().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

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with LMFDB labels.