Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("51870.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 51870.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
51870.e1 51870b4 [1, 1, 0, -3688533, -2728182627] [2] 655360  
51870.e2 51870b3 [1, 1, 0, -233653, -41488643] [2] 655360  
51870.e3 51870b2 [1, 1, 0, -230533, -42699827] [2, 2] 327680  
51870.e4 51870b1 [1, 1, 0, -14213, -690483] [2] 163840 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Modular form 51870.2.a.e

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

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 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.