Properties

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

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Show commands for: SageMath

sage: E = EllipticCurve("51870.i1")
sage: E.isogeny_class()

Elliptic curves in class 51870h

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.i3 51870h1 [1, 1, 0, -461968, 120663088] 2 294912 \(\Gamma_0(N)\)-optimal
51870.i2 51870h2 [1, 1, 0, -461988, 120652092] 4 589824  
51870.i4 51870h3 [1, 1, 0, -415058, 146191398] 2 1179648  
51870.i1 51870h4 [1, 1, 0, -509238, 94409442] 2 1179648  

Rank

sage: E.rank()

The elliptic curves in class 51870h 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.