Properties

Label 862.d
Number of curves 2
Conductor 862
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("862.d1")
sage: E.isogeny_class()

Elliptic curves in class 862.d

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
862.d1 862e1 [1, 1, 1, -2460, 45949] 5 640 \(\Gamma_0(N)\)-optimal
862.d2 862e2 [1, 1, 1, 15380, -102531] 1 3200  

Rank

sage: E.rank()

The elliptic curves in class 862.d have rank \(1\).

Modular form 862.2.a.d

sage: E.q_eigenform(10)
\( q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2q^{7} + q^{8} - 2q^{9} + q^{10} - 3q^{11} - q^{12} - 6q^{13} - 2q^{14} - q^{15} + q^{16} - 2q^{17} - 2q^{18} - 5q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.