Properties

Label 51870.b
Number of curves 4
Conductor 51870
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("51870.b1")
sage: E.isogeny_class()

Elliptic curves in class 51870.b

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.b1 51870e4 [1, 1, 0, -428140743, -3409726086027] 2 19464192  
51870.b2 51870e3 [1, 1, 0, -153375943, 692540686453] 2 19464192  
51870.b3 51870e2 [1, 1, 0, -28614343, -45474082187] 4 9732096  
51870.b4 51870e1 [1, 1, 0, 4153657, -4428885387] 2 4866048 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()

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

Modular form None

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

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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.