Properties

Label 50.a
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 50.a

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

Rank

sage: E.rank()
 

The elliptic curves in class 50.a 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.