Properties

Label 5082i
Number of curves $6$
Conductor $5082$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("5082.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 5082i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
5082.d5 5082i1 [1, 1, 0, -486, -9324] [2] 5120 \(\Gamma_0(N)\)-optimal
5082.d4 5082i2 [1, 1, 0, -10166, -398460] [2, 2] 10240  
5082.d1 5082i3 [1, 1, 0, -162626, -25310424] [2] 20480  
5082.d3 5082i4 [1, 1, 0, -12586, -197600] [2, 2] 20480  
5082.d2 5082i5 [1, 1, 0, -110596, 13974646] [2] 40960  
5082.d6 5082i6 [1, 1, 0, 46704, -1466406] [2] 40960  

Rank

sage: E.rank()
 

The elliptic curves in class 5082i have rank \(1\).

Modular form 5082.2.a.d

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.