Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("50.b1")
sage: E.isogeny_class()

Elliptic curves in class 50.b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
50.b1 50b4 [1, 1, 1, -3138, -68969] 1 30  
50.b2 50b3 [1, 1, 1, -13, -219] 1 10  
50.b3 50b1 [1, 1, 1, -3, 1] 5 2 \(\Gamma_0(N)\)-optimal
50.b4 50b2 [1, 1, 1, 22, -9] 5 6  

Rank

sage: E.rank()

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

Modular form 50.2.a.b

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.