# Properties

 Label 55545h Number of curves 2 Conductor 55545 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55545.d1")

sage: E.isogeny_class()

## Elliptic curves in class 55545h

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55545.d1 55545h1 [1, 0, 0, -121681, 14028536]  405504 $$\Gamma_0(N)$$-optimal
55545.d2 55545h2 [1, 0, 0, 208944, 77045661]  811008

## Rank

sage: E.rank()

The elliptic curves in class 55545h have rank $$1$$.

## Modular form 55545.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} + q^{10} + 2q^{11} - q^{12} - 4q^{13} + q^{14} - q^{15} - q^{16} + 2q^{17} - q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 