Properties

Label 51870r
Number of curves 2
Conductor 51870
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("51870.r1")
sage: E.isogeny_class()

Elliptic curves in class 51870r

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.r1 51870r1 [1, 1, 0, -72, 204] 2 10240 \(\Gamma_0(N)\)-optimal
51870.r2 51870r2 [1, 1, 0, -2, 666] 2 20480  

Rank

sage: E.rank()

The elliptic curves in class 51870r have rank \(1\).

Modular form None

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

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.