Properties

Label 286650.kf
Number of curves $6$
Conductor $286650$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("286650.kf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 286650.kf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
286650.kf1 286650kf6 [1, -1, 1, -99390605, -381362306853] [2] 37748736  
286650.kf2 286650kf3 [1, -1, 1, -9316355, 10938724647] [2] 18874368  
286650.kf3 286650kf4 [1, -1, 1, -6229355, -5922469353] [2, 2] 18874368  
286650.kf4 286650kf5 [1, -1, 1, -1268105, -15100781853] [2] 37748736  
286650.kf5 286650kf2 [1, -1, 1, -716855, 86155647] [2, 2] 9437184  
286650.kf6 286650kf1 [1, -1, 1, 165145, 10303647] [2] 4718592 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 286650.kf have rank \(1\).

Modular form 286650.2.a.kf

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + q^{8} - 4q^{11} + q^{13} + q^{16} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.