Properties

Label 221760.p
Number of curves 4
Conductor 221760
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("221760.p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 221760.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221760.p1 221760lz4 [0, 0, 0, -19002828, -31883804048] [2] 9437184  
221760.p2 221760lz2 [0, 0, 0, -1221708, -468121232] [2, 2] 4718592  
221760.p3 221760lz1 [0, 0, 0, -288588, 51813232] [2] 2359296 \(\Gamma_0(N)\)-optimal
221760.p4 221760lz3 [0, 0, 0, 1629492, -2328244112] [2] 9437184  

Rank

sage: E.rank()
 

The elliptic curves in class 221760.p have rank \(2\).

Modular form 221760.2.a.p

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.