# Properties

 Label 19890e Number of curves 2 Conductor 19890 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("19890.e1")

sage: E.isogeny_class()

## Elliptic curves in class 19890e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
19890.e1 19890e1 [1, -1, 0, -122580, -16299824]  122880 $$\Gamma_0(N)$$-optimal
19890.e2 19890e2 [1, -1, 0, -18900, -43070000]  245760

## Rank

sage: E.rank()

The elliptic curves in class 19890e have rank $$1$$.

## Modular form 19890.2.a.e

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} - 2q^{7} - q^{8} + q^{10} - q^{13} + 2q^{14} + q^{16} + q^{17} + 2q^{19} + O(q^{20})$$

## 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 Cremona 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 Cremona labels. 