Properties

Label 10005j
Number of curves 2
Conductor 10005
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("10005.m1")
sage: E.isogeny_class()

Elliptic curves in class 10005j

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
10005.m1 10005j1 [1, 0, 1, -374, 2747] 2 1728 \(\Gamma_0(N)\)-optimal
10005.m2 10005j2 [1, 0, 1, -349, 3137] 2 3456  

Rank

sage: E.rank()

The elliptic curves in class 10005j have rank \(1\).

Modular form 10005.2.a.m

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

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 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.