Properties

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

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 50a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
50.a3 50a1 \([1, 0, 1, -1, -2]\) \(-25/2\) \(-1250\) \([3]\) \(2\) \(-0.72610\) \(\Gamma_0(N)\)-optimal
50.a1 50a2 \([1, 0, 1, -126, -552]\) \(-349938025/8\) \(-5000\) \([]\) \(6\) \(-0.17679\)  
50.a2 50a3 \([1, 0, 1, -76, 298]\) \(-121945/32\) \(-12500000\) \([3]\) \(10\) \(0.078619\)  
50.a4 50a4 \([1, 0, 1, 549, -2202]\) \(46969655/32768\) \(-12800000000\) \([]\) \(30\) \(0.62793\)  

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 50a do not have complex multiplication.

Modular form 50.2.a.a

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

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.