Properties

Label 86490cw
Number of curves $4$
Conductor $86490$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cw1")
 
E.isogeny_class()
 

Elliptic curves in class 86490cw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86490.cx3 86490cw1 \([1, -1, 1, -2106212, 1176933111]\) \(1597099875769/186000\) \(120340174121514000\) \([4]\) \(2211840\) \(2.3037\) \(\Gamma_0(N)\)-optimal
86490.cx2 86490cw2 \([1, -1, 1, -2279192, 972401559]\) \(2023804595449/540562500\) \(349738631040650062500\) \([2, 2]\) \(4423680\) \(2.6503\)  
86490.cx4 86490cw3 \([1, -1, 1, 5764378, 6294027471]\) \(32740359775271/45410156250\) \(-29379925322635253906250\) \([2]\) \(8847360\) \(2.9969\)  
86490.cx1 86490cw4 \([1, -1, 1, -13090442, -17441319441]\) \(383432500775449/18701300250\) \(12099557679482329562250\) \([2]\) \(8847360\) \(2.9969\)  

Rank

sage: E.rank()
 

The elliptic curves in class 86490cw have rank \(1\).

Complex multiplication

The elliptic curves in class 86490cw do not have complex multiplication.

Modular form 86490.2.a.cw

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.