# Properties

 Label 219351.f Number of curves 2 Conductor 219351 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("219351.f1")

sage: E.isogeny_class()

## Elliptic curves in class 219351.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
219351.f1 219351d2 [1, 0, 0, -357788, -81762387]  1658880
219351.f2 219351d1 [1, 0, 0, -6653, -3037920]  829440 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 219351.f have rank $$0$$.

## Modular form 219351.2.a.f

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 