# Properties

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

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 256.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
256.a1 256a2 $$[0, 1, 0, -13, -21]$$ $$8000$$ $$32768$$ $$$$ $$16$$ $$-0.40397$$   $$-8$$
256.a2 256a1 $$[0, 1, 0, -3, 1]$$ $$8000$$ $$512$$ $$$$ $$8$$ $$-0.75055$$ $$\Gamma_0(N)$$-optimal $$-8$$

## Rank

sage: E.rank()

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

## Complex multiplication

Each elliptic curve in class 256.a has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-2})$$.

## Modular form256.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{3} + q^{9} - 6q^{11} - 6q^{17} - 2q^{19} + O(q^{20})$$ ## 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. 