Properties

Label 66759g
Number of curves 6
Conductor 66759
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("66759.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 66759g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
66759.e4 66759g1 [1, 0, 0, -9832, 374303] [2] 102400 \(\Gamma_0(N)\)-optimal
66759.e3 66759g2 [1, 0, 0, -11277, 256680] [2, 2] 204800  
66759.e6 66759g3 [1, 0, 0, 36408, 1868433] [2] 409600  
66759.e2 66759g4 [1, 0, 0, -82082, -8877165] [2, 2] 409600  
66759.e5 66759g5 [1, 0, 0, 8953, -27466512] [2] 819200  
66759.e1 66759g6 [1, 0, 0, -1305997, -574570678] [2] 819200  

Rank

sage: E.rank()
 

The elliptic curves in class 66759g have rank \(1\).

Modular form 66759.2.a.e

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} - 2q^{10} + q^{11} - q^{12} + 6q^{13} + q^{14} + 2q^{15} - q^{16} - q^{18} + 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.