Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("289578.a1")
sage: E.isogeny_class()

Elliptic curves in class 289578a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
289578.a2 289578a1 [1, 1, 0, -1306, -76076] 2 709632 \(\Gamma_0(N)\)-optimal*
289578.a1 289578a2 [1, 1, 0, -35986, -2635460] 2 1419264 \(\Gamma_0(N)\)-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 289578a1.

Rank

sage: E.rank()

The elliptic curves in class 289578a 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 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.