# Properties

 Label 55.a Number of curves 4 Conductor 55 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55.a1")
sage: E.isogeny_class()

## Elliptic curves in class 55.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
55.a1 55a3 [1, -1, 0, -59, 190] 4 4
55.a2 55a2 [1, -1, 0, -29, -52] 2 4
55.a3 55a1 [1, -1, 0, -4, 3] 4 2 $$\Gamma_0(N)$$-optimal
55.a4 55a4 [1, -1, 0, 1, 0] 2 4

## Rank

sage: E.rank()

The elliptic curves in class 55.a have rank $$0$$.

## Modular form55.2.a.a

sage: E.q_eigenform(10)
$$q + q^{2} - q^{4} + q^{5} - 3q^{8} - 3q^{9} + q^{10} - q^{11} + 2q^{13} - q^{16} + 6q^{17} - 3q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 