# Properties

 Label 289578.a Number of curves 2 Conductor 289578 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("289578.a1")
sage: E.isogeny_class()

## Elliptic curves in class 289578.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
289578.a1 289578a2 [1, 1, 0, -35986, -2635460] 2 1419264 $$\Gamma_0(N)$$-optimal*
289578.a2 289578a1 [1, 1, 0, -1306, -76076] 2 709632 $$\Gamma_0(N)$$-optimal*
*optimality has not been proved rigorously for conductors over 270000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 289578.a2.

## Rank

sage: E.rank()

The elliptic curves in class 289578.a have rank $$1$$.

## Modular form None

sage: E.q_eigenform(10)
$$q - q^{2} - q^{3} + q^{4} - 2q^{5} + q^{6} + 4q^{7} - q^{8} + q^{9} + 2q^{10} + 4q^{11} - q^{12} - 4q^{14} + 2q^{15} + q^{16} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.