Properties

Label 51870.d
Number of curves 4
Conductor 51870
CM no
Rank 2
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("51870.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51870.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.d1 51870a4 [1, 1, 0, -108528, 13715688] [2] 245760  
51870.d2 51870a3 [1, 1, 0, -33728, -2221032] [2] 245760  
51870.d3 51870a2 [1, 1, 0, -7128, 188928] [2, 2] 122880  
51870.d4 51870a1 [1, 1, 0, 872, 17728] [2] 61440 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 51870.d have rank \(2\).

Modular form 51870.2.a.d

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

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.