Properties

Label 159936fx
Number of curves 6
Conductor 159936
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("159936.gx1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 159936fx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
159936.gx5 159936fx1 [0, 1, 0, -106689, -12475233] [2] 1179648 \(\Gamma_0(N)\)-optimal
159936.gx4 159936fx2 [0, 1, 0, -357569, 67756191] [2, 2] 2359296  
159936.gx2 159936fx3 [0, 1, 0, -5437889, 4878819231] [2, 2] 4718592  
159936.gx6 159936fx4 [0, 1, 0, 708671, 395518367] [2] 4718592  
159936.gx1 159936fx5 [0, 1, 0, -87005249, 312338826015] [2] 9437184  
159936.gx3 159936fx6 [0, 1, 0, -5155649, 5408132127] [2] 9437184  

Rank

sage: E.rank()
 

The elliptic curves in class 159936fx have rank \(1\).

Modular form 159936.2.a.gx

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

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.