Properties

Label 129360du
Number of curves $6$
Conductor $129360$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("129360.w1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 129360du

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
129360.w4 129360du1 [0, -1, 0, -207776, 36522816] [2] 786432 \(\Gamma_0(N)\)-optimal
129360.w3 129360du2 [0, -1, 0, -211696, 35077120] [2, 2] 1572864  
129360.w5 129360du3 [0, -1, 0, 199904, 154605760] [2] 3145728  
129360.w2 129360du4 [0, -1, 0, -686016, -177038784] [2, 2] 3145728  
129360.w6 129360du5 [0, -1, 0, 1426864, -1054306560] [2] 6291456  
129360.w1 129360du6 [0, -1, 0, -10388016, -12882777984] [2] 6291456  

Rank

sage: E.rank()
 

The elliptic curves in class 129360du have rank \(1\).

Modular form 129360.2.a.w

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

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.