# Properties

 Label 441f Number of curves 2 Conductor 441 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("441.a1")

sage: E.isogeny_class()

## Elliptic curves in class 441f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
441.a2 441f1 [0, 0, 1, -21, 40] [] 48 $$\Gamma_0(N)$$-optimal
441.a1 441f2 [0, 0, 1, -8211, -286610] [] 624

## Rank

sage: E.rank()

The elliptic curves in class 441f have rank $$1$$.

## Modular form441.2.a.a

sage: E.q_eigenform(10)

$$q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{10} + 2q^{11} - q^{13} - 4q^{16} - q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 