Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("51870.n1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51870o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.n4 51870o1 [1, 1, 0, 838, 1991316] [2] 368640 \(\Gamma_0(N)\)-optimal
51870.n3 51870o2 [1, 1, 0, -232442, 42255444] [2, 2] 737280  
51870.n2 51870o3 [1, 1, 0, -497042, -72104676] [2] 1474560  
51870.n1 51870o4 [1, 1, 0, -3700322, 2738185356] [2] 1474560  

Rank

sage: E.rank()
 

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

Modular form 51870.2.a.n

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

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.