Properties

Label 361722.u
Number of curves 2
Conductor 361722
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("361722.u1")
sage: E.isogeny_class()

Elliptic curves in class 361722.u

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
361722.u1 361722u2 [1, 1, 1, -44952, 3639681] 2 1990656 \(\Gamma_0(N)\)-optimal*
361722.u2 361722u1 [1, 1, 1, -1632, 104769] 2 995328 \(\Gamma_0(N)\)-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 361722.u2.

Rank

sage: E.rank()

The elliptic curves in class 361722.u have rank \(1\).

Modular form None

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

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.