Properties

Label 68208.be
Number of curves $6$
Conductor $68208$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("68208.be1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 68208.be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
68208.be1 68208bw4 [0, -1, 0, -160425232, 782144188288] [2] 4718592  
68208.be2 68208bw6 [0, -1, 0, -33295712, -60103232832] [2] 9437184  
68208.be3 68208bw3 [0, -1, 0, -10218672, 11730977280] [2, 2] 4718592  
68208.be4 68208bw2 [0, -1, 0, -10026592, 12223470400] [2, 2] 2359296  
68208.be5 68208bw1 [0, -1, 0, -614672, 198801408] [2] 1179648 \(\Gamma_0(N)\)-optimal
68208.be6 68208bw5 [0, -1, 0, 9785088, 52042554432] [4] 9437184  

Rank

sage: E.rank()
 

The elliptic curves in class 68208.be have rank \(0\).

Modular form 68208.2.a.be

sage: E.q_eigenform(10)
 
\( q - q^{3} + 2q^{5} + q^{9} - 4q^{11} + 2q^{13} - 2q^{15} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.