Properties

Label 221760cb
Number of curves $6$
Conductor $221760$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cb1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 221760cb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221760.jp6 221760cb1 [0, 0, 0, 20148, -319761776] [2] 3932160 \(\Gamma_0(N)\)-optimal
221760.jp5 221760cb2 [0, 0, 0, -6894732, -6844642544] [2, 2] 7864320  
221760.jp4 221760cb3 [0, 0, 0, -14656332, 11388908176] [2] 15728640  
221760.jp2 221760cb4 [0, 0, 0, -109771212, -442670562416] [2, 2] 15728640  
221760.jp3 221760cb5 [0, 0, 0, -109226892, -447277904624] [2] 31457280  
221760.jp1 221760cb6 [0, 0, 0, -1756339212, -28330922092016] [2] 31457280  

Rank

sage: E.rank()
 

The elliptic curves in class 221760cb have rank \(1\).

Complex multiplication

The elliptic curves in class 221760cb do not have complex multiplication.

Modular form 221760.2.a.cb

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.