Properties

Label 39270bh
Number of curves 8
Conductor 39270
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("39270.bg1")
sage: E.isogeny_class()

Elliptic curves in class 39270bh

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
39270.bg7 39270bh1 [1, 0, 1, -945654, -2966648] 6 1327104 \(\Gamma_0(N)\)-optimal
39270.bg5 39270bh2 [1, 0, 1, -10357574, 12789715016] 12 2654208  
39270.bg4 39270bh3 [1, 0, 1, -52843269, -147857449424] 2 3981312  
39270.bg6 39270bh4 [1, 0, 1, -5727074, 24284468216] 6 5308416  
39270.bg2 39270bh5 [1, 0, 1, -165578794, 820064235992] 6 5308416  
39270.bg3 39270bh6 [1, 0, 1, -53846789, -141949927888] 4 7962624  
39270.bg8 39270bh7 [1, 0, 1, 51153211, -628351927888] 2 15925248  
39270.bg1 39270bh8 [1, 0, 1, -174903109, 722537464496] 2 15925248  

Rank

sage: E.rank()

The elliptic curves in class 39270bh have rank \(0\).

Modular form None

sage: E.q_eigenform(10)
\( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + 2q^{13} - q^{14} - q^{15} + q^{16} - q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.