# Properties

 Label 550.f Number of curves $2$ Conductor $550$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 550.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
550.f1 550b2 $$[1, 0, 1, -148501, -22038602]$$ $$-23178622194826561/1610510$$ $$-25164218750$$ $$[]$$ $$2400$$ $$1.4498$$
550.f2 550b1 $$[1, 0, 1, 249, -6102]$$ $$109902239/1100000$$ $$-17187500000$$ $$[]$$ $$480$$ $$0.64508$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 550.f have rank $$0$$.

## Complex multiplication

The elliptic curves in class 550.f do not have complex multiplication.

## Modular form550.2.a.f

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{6} - 3 q^{7} - q^{8} - 2 q^{9} + q^{11} + q^{12} + 6 q^{13} + 3 q^{14} + q^{16} + 7 q^{17} + 2 q^{18} + 5 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 