Properties

Label 15870bg
Number of curves $6$
Conductor $15870$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("15870.bg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 15870bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
15870.bg5 15870bg1 [1, 0, 0, -222191, -44245575] [4] 202752 \(\Gamma_0(N)\)-optimal
15870.bg4 15870bg2 [1, 0, 0, -3650111, -2684429559] [2, 2] 405504  
15870.bg1 15870bg3 [1, 0, 0, -58401611, -171789912459] [2] 811008  
15870.bg3 15870bg4 [1, 0, 0, -3745331, -2537009955] [2, 2] 811008  
15870.bg2 15870bg5 [1, 0, 0, -13664081, 16651803795] [2] 1622016  
15870.bg6 15870bg6 [1, 0, 0, 4649899, -12290588169] [2] 1622016  

Rank

sage: E.rank()
 

The elliptic curves in class 15870bg have rank \(0\).

Modular form 15870.2.a.bg

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.