# Properties

 Label 55545u Number of curves 4 Conductor 55545 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55545.g1")

sage: E.isogeny_class()

## Elliptic curves in class 55545u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55545.g4 55545u1 [1, 0, 0, -122210, -4332525]  557568 $$\Gamma_0(N)$$-optimal
55545.g2 55545u2 [1, 0, 0, -1521415, -721565008] [2, 2] 1115136
55545.g3 55545u3 [1, 0, 0, -1095570, -1134038475]  2230272
55545.g1 55545u4 [1, 0, 0, -24334540, -46206373633]  2230272

## Rank

sage: E.rank()

The elliptic curves in class 55545u have rank $$0$$.

## Modular form 55545.2.a.g

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 