Properties

Label 105966bu
Number of curves $6$
Conductor $105966$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("105966.ce1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 105966bu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
105966.ce5 105966bu1 [1, -1, 1, -30434, -4505119] [2] 802816 \(\Gamma_0(N)\)-optimal
105966.ce4 105966bu2 [1, -1, 1, -635954, -194880607] [2, 2] 1605632  
105966.ce3 105966bu3 [1, -1, 1, -787334, -94969807] [2, 2] 3211264  
105966.ce1 105966bu4 [1, -1, 1, -10172894, -12486088879] [2] 3211264  
105966.ce6 105966bu5 [1, -1, 1, 2921476, -735852175] [2] 6422528  
105966.ce2 105966bu6 [1, -1, 1, -6918224, 6938387201] [2] 6422528  

Rank

sage: E.rank()
 

The elliptic curves in class 105966bu have rank \(1\).

Modular form 105966.2.a.ce

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + 2q^{5} - q^{7} + q^{8} + 2q^{10} - 4q^{11} + 6q^{13} - q^{14} + q^{16} + 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.