Properties

Label 123.a
Number of curves 2
Conductor 123
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("123.a1")
sage: E.isogeny_class()

Elliptic curves in class 123.a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
123.a1 123a1 [0, 1, 1, -10, 10] 5 20 \(\Gamma_0(N)\)-optimal
123.a2 123a2 [0, 1, 1, 20, -890] 1 100  

Rank

sage: E.rank()

The elliptic curves in class 123.a have rank \(1\).

Modular form 123.2.a.a

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

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.