Properties

Label 106722cu
Number of curves 4
Conductor 106722
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("106722.bu1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 106722cu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
106722.bu3 106722cu1 [1, -1, 0, -294597, -57977375] [2] 1382400 \(\Gamma_0(N)\)-optimal
106722.bu4 106722cu2 [1, -1, 0, 239013, -245061041] [2] 2764800  
106722.bu1 106722cu3 [1, -1, 0, -4296672, 3415983808] [2] 4147200  
106722.bu2 106722cu4 [1, -1, 0, -2162232, 6809316520] [2] 8294400  

Rank

sage: E.rank()
 

The elliptic curves in class 106722cu have rank \(2\).

Modular form 106722.2.a.bu

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.