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Properties

Label 5082e
Number of curves $2$
Conductor $5082$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 5082e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5082.a2 5082e1 \([1, 1, 0, -49007, -6542235]\) \(-7347774183121/6119866368\) \(-10841716582760448\) \([2]\) \(80640\) \(1.7747\) \(\Gamma_0(N)\)-optimal
5082.a1 5082e2 \([1, 1, 0, -900847, -329389595]\) \(45637459887836881/13417633152\) \(23770155604390272\) \([2]\) \(161280\) \(2.1213\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5082e have rank \(0\).

Complex multiplication

The elliptic curves in class 5082e do not have complex multiplication.

Modular form 5082.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 4 q^{5} + q^{6} - q^{7} - q^{8} + q^{9} + 4 q^{10} - q^{12} + 6 q^{13} + q^{14} + 4 q^{15} + q^{16} + 4 q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.