Properties

Label 51870.y
Number of curves 8
Conductor 51870
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("51870.y1")
sage: E.isogeny_class()

Elliptic curves in class 51870.y

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.y1 51870bd8 [1, 0, 1, -768208004, -8195392298998] 2 7962624  
51870.y2 51870bd6 [1, 0, 1, -48013004, -128055986998] 4 3981312  
51870.y3 51870bd7 [1, 0, 1, -47818004, -129147674998] 2 7962624  
51870.y4 51870bd5 [1, 0, 1, -9485039, -11240205244] 6 2654208  
51870.y5 51870bd3 [1, 0, 1, -3013004, -1983986998] 2 1990656  
51870.y6 51870bd2 [1, 0, 1, -687089, -116077264] 12 1327104  
51870.y7 51870bd1 [1, 0, 1, -322589, 69234536] 6 663552 \(\Gamma_0(N)\)-optimal
51870.y8 51870bd4 [1, 0, 1, 2278861, -850446484] 6 2654208  

Rank

sage: E.rank()

The elliptic curves in class 51870.y have rank \(1\).

Modular form 51870.2.a.y

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

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.