Properties

Label 66270m
Number of curves 8
Conductor 66270
CM no
Rank 0
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("66270.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 66270m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
66270.k8 66270m1 [1, 0, 1, 3267, -220472] [2] 211968 \(\Gamma_0(N)\)-optimal
66270.k6 66270m2 [1, 0, 1, -40913, -2888944] [2, 2] 423936  
66270.k7 66270m3 [1, 0, 1, -29868, 6499306] [2] 635904  
66270.k5 66270m4 [1, 0, 1, -151363, 19510316] [2] 847872  
66270.k4 66270m5 [1, 0, 1, -637343, -195893692] [2] 847872  
66270.k3 66270m6 [1, 0, 1, -736748, 242879978] [2, 2] 1271808  
66270.k1 66270m7 [1, 0, 1, -11781748, 15564503978] [2] 2543616  
66270.k2 66270m8 [1, 0, 1, -1001828, 52446506] [2] 2543616  

Rank

sage: E.rank()
 

The elliptic curves in class 66270m have rank \(0\).

Modular form 66270.2.a.k

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

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 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.