Properties

Label 221760kw
Number of curves $4$
Conductor $221760$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("kw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 221760kw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221760.dv3 221760kw1 \([0, 0, 0, -95808, -11583448]\) \(-130287139815424/2250652635\) \(-1680103189416960\) \([2]\) \(1327104\) \(1.7194\) \(\Gamma_0(N)\)-optimal
221760.dv2 221760kw2 \([0, 0, 0, -1539228, -735025552]\) \(33766427105425744/9823275\) \(117328567910400\) \([2]\) \(2654208\) \(2.0660\)  
221760.dv4 221760kw3 \([0, 0, 0, 370752, -55510072]\) \(7549996227362816/6152409907875\) \(-4592749386589056000\) \([2]\) \(3981312\) \(2.2687\)  
221760.dv1 221760kw4 \([0, 0, 0, -1785468, -484166608]\) \(52702650535889104/22020583921875\) \(263012445045504000000\) \([2]\) \(7962624\) \(2.6153\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 221760.2.a.kw

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

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.