Properties

Label 163170de
Number of curves $6$
Conductor $163170$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("163170.ca1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 163170de

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
163170.ca5 163170de1 [1, -1, 0, -372654, 301513428] [2] 5308416 \(\Gamma_0(N)\)-optimal
163170.ca4 163170de2 [1, -1, 0, -9404334, 11079920340] [2, 2] 10616832  
163170.ca1 163170de3 [1, -1, 0, -150383214, 709855836948] [2] 21233664  
163170.ca3 163170de4 [1, -1, 0, -12932334, 2013665940] [2, 2] 21233664  
163170.ca6 163170de5 [1, -1, 0, 51365466, 16017726780] [2] 42467328  
163170.ca2 163170de6 [1, -1, 0, -133678134, -592273012500] [2] 42467328  

Rank

sage: E.rank()
 

The elliptic curves in class 163170de have rank \(1\).

Modular form 163170.2.a.ca

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

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.