# Properties

 Label 56a Number of curves 4 Conductor 56 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("56.a1")

sage: E.isogeny_class()

## Elliptic curves in class 56a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
56.a4 56a1 [0, 0, 0, 1, 2]  2 $$\Gamma_0(N)$$-optimal
56.a3 56a2 [0, 0, 0, -19, 30] [2, 2] 4
56.a2 56a3 [0, 0, 0, -59, -138]  8
56.a1 56a4 [0, 0, 0, -299, 1990]  8

## Rank

sage: E.rank()

The elliptic curves in class 56a have rank $$0$$.

## Modular form56.2.a.a

sage: E.q_eigenform(10)

$$q + 2q^{5} - q^{7} - 3q^{9} - 4q^{11} + 2q^{13} - 6q^{17} + 8q^{19} + O(q^{20})$$

## 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 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

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

The vertices are labelled with Cremona labels. 