Properties

Label 330330.cr
Number of curves 8
Conductor 330330
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("330330.cr1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 330330.cr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
330330.cr1 330330cr7 [1, 0, 1, -4863045058, 130529659590878] [2] 159252480  
330330.cr2 330330cr6 [1, 0, 1, -303940508, 2039504237318] [2, 2] 79626240  
330330.cr3 330330cr8 [1, 0, 1, -300205238, 2092075921406] [2] 159252480  
330330.cr4 330330cr4 [1, 0, 1, -60065008, 178876879718] [2] 53084160  
330330.cr5 330330cr3 [1, 0, 1, -19229928, 31041921766] [2] 39813120  
330330.cr6 330330cr2 [1, 0, 1, -5010008, 762943718] [2, 2] 26542080  
330330.cr7 330330cr1 [1, 0, 1, -3112728, -2102707994] [2] 13271040 \(\Gamma_0(N)\)-optimal
330330.cr8 330330cr5 [1, 0, 1, 19688512, 6058306406] [2] 53084160  

Rank

sage: E.rank()
 

The elliptic curves in class 330330.cr have rank \(2\).

Modular form 330330.2.a.cr

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.