# Properties

 Label 55545.p Number of curves 2 Conductor 55545 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("55545.p1")

sage: E.isogeny_class()

## Elliptic curves in class 55545.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55545.p1 55545r1 [0, 1, 1, -11285, -1431244] [] 228096 $$\Gamma_0(N)$$-optimal
55545.p2 55545r2 [0, 1, 1, 99805, 34972949] [] 684288

## Rank

sage: E.rank()

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

## Modular form 55545.2.a.p

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{4} + q^{5} - q^{7} + q^{9} - 3q^{11} - 2q^{12} - 4q^{13} + q^{15} + 4q^{16} + 6q^{17} + q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 