Properties

Label 136710du
Number of curves $6$
Conductor $136710$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("136710.dd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 136710du

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
136710.dd6 136710du1 [1, -1, 0, 26451, -9387995] [2] 1572864 \(\Gamma_0(N)\)-optimal
136710.dd5 136710du2 [1, -1, 0, -538029, -143395547] [2, 2] 3145728  
136710.dd4 136710du3 [1, -1, 0, -1631709, 625898965] [2] 6291456  
136710.dd2 136710du4 [1, -1, 0, -8476029, -9495947147] [2, 2] 6291456  
136710.dd3 136710du5 [1, -1, 0, -8343729, -9806825687] [2] 12582912  
136710.dd1 136710du6 [1, -1, 0, -135616329, -607843627007] [2] 12582912  

Rank

sage: E.rank()
 

The elliptic curves in class 136710du have rank \(1\).

Modular form 136710.2.a.dd

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

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.