# Properties

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

Show commands for: SageMath
sage: E = EllipticCurve("55545.f1")

sage: E.isogeny_class()

## Elliptic curves in class 55545.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55545.f1 55545m4 [1, 0, 0, -2343481, 1376328380]  1081344
55545.f2 55545m2 [1, 0, 0, -214256, -428505] [2, 2] 540672
55545.f3 55545m1 [1, 0, 0, -148131, -21892680]  270336 $$\Gamma_0(N)$$-optimal
55545.f4 55545m3 [1, 0, 0, 856969, -3213690]  1081344

## Rank

sage: E.rank()

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

## Modular form 55545.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + q^{7} + 3q^{8} + q^{9} + q^{10} - q^{12} + 2q^{13} - q^{14} - q^{15} - q^{16} + 6q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 