Properties

Label 86640bw
Number of curves $4$
Conductor $86640$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 86640bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86640.j3 86640bw1 \([0, -1, 0, -179176, 29077360]\) \(3301293169/22800\) \(4393558371532800\) \([2]\) \(552960\) \(1.8351\) \(\Gamma_0(N)\)-optimal
86640.j2 86640bw2 \([0, -1, 0, -294696, -12879504]\) \(14688124849/8122500\) \(1565205169858560000\) \([2, 2]\) \(1105920\) \(2.1817\)  
86640.j4 86640bw3 \([0, -1, 0, 1149304, -102985104]\) \(871257511151/527800050\) \(-101707031937409228800\) \([2]\) \(2211840\) \(2.5283\)  
86640.j1 86640bw4 \([0, -1, 0, -3587016, -2609861520]\) \(26487576322129/44531250\) \(8581168694400000000\) \([2]\) \(2211840\) \(2.5283\)  

Rank

sage: E.rank()
 

The elliptic curves in class 86640bw have rank \(1\).

Complex multiplication

The elliptic curves in class 86640bw do not have complex multiplication.

Modular form 86640.2.a.bw

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} + q^{9} - 4q^{11} - 2q^{13} + q^{15} + 2q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.