Properties

Label 6630w
Number of curves 8
Conductor 6630
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("6630.v1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6630w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6630.v7 6630w1 [1, 0, 0, -2096146, 1124611076] [12] 276480 \(\Gamma_0(N)\)-optimal
6630.v6 6630w2 [1, 0, 0, -5557266, -3547208700] [2, 6] 552960  
6630.v5 6630w3 [1, 0, 0, -25792786, -50092914940] [4] 829440  
6630.v4 6630w4 [1, 0, 0, -81373266, -282504599100] [6] 1105920  
6630.v8 6630w5 [1, 0, 0, 14880814, -23572439484] [6] 1105920  
6630.v2 6630w6 [1, 0, 0, -411937506, -3218101426764] [2, 2] 1658880  
6630.v1 6630w7 [1, 0, 0, -6591000006, -205956849489264] [2] 3317760  
6630.v3 6630w8 [1, 0, 0, -411190526, -3230353542120] [2] 3317760  

Rank

sage: E.rank()
 

The elliptic curves in class 6630w have rank \(1\).

Modular form 6630.2.a.v

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} - 4q^{7} + q^{8} + q^{9} - q^{10} + q^{12} + q^{13} - 4q^{14} - q^{15} + q^{16} + q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.