Properties

Label 50a
Number of curves 4
Conductor 50
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("50.a1")
sage: E.isogeny_class()

Elliptic curves in class 50a

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
50.a3 50a1 [1, 0, 1, -1, -2] 3 2 \(\Gamma_0(N)\)-optimal
50.a1 50a2 [1, 0, 1, -126, -552] 1 6  
50.a2 50a3 [1, 0, 1, -76, 298] 3 10  
50.a4 50a4 [1, 0, 1, 549, -2202] 1 30  

Rank

sage: E.rank()

The elliptic curves in class 50a have rank \(0\).

Modular form 50.2.a.a

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

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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.