Properties

Label 51870e
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 51870e

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

Rank

sage: E.rank()
 

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

Modular form 51870.2.a.b

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.