Properties

Label 277200.gy
Number of curves 6
Conductor 277200
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("277200.gy1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 277200.gy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
277200.gy1 277200gy6 [0, 0, 0, -16268475, -25256209750] [2] 10485760  
277200.gy2 277200gy4 [0, 0, 0, -1022475, -389983750] [2, 2] 5242880  
277200.gy3 277200gy2 [0, 0, 0, -140475, 11326250] [2, 2] 2621440  
277200.gy4 277200gy1 [0, 0, 0, -122475, 16492250] [2] 1310720 \(\Gamma_0(N)\)-optimal
277200.gy5 277200gy5 [0, 0, 0, 111525, -1207597750] [2] 10485760  
277200.gy6 277200gy3 [0, 0, 0, 453525, 82012250] [2] 5242880  

Rank

sage: E.rank()
 

The elliptic curves in class 277200.gy have rank \(2\).

Modular form 277200.2.a.gy

sage: E.q_eigenform(10)
 
\( q + q^{7} - q^{11} - 6q^{13} + 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.