Properties

Label 51870j
Number of curves 4
Conductor 51870
CM no
Rank 0
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("51870.j1")
sage: E.isogeny_class()

Elliptic curves in class 51870j

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
51870.j4 51870j1 [1, 1, 0, -69218, -2015628] 2 589824 \(\Gamma_0(N)\)-optimal
51870.j2 51870j2 [1, 1, 0, -869218, -311935628] 4 1179648  
51870.j3 51870j3 [1, 1, 0, -635218, -483364028] 4 2359296  
51870.j1 51870j4 [1, 1, 0, -13903218, -19959387228] 2 2359296  

Rank

sage: E.rank()

The elliptic curves in class 51870j have rank \(0\).

Modular form None

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

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.