Properties

Label 491970.ci
Number of curves $6$
Conductor $491970$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("491970.ci1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 491970.ci

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
491970.ci1 491970ci5 [1, 0, 0, -162678091, 798609261845] [2] 46137344 \(\Gamma_0(N)\)-optimal*
491970.ci2 491970ci3 [1, 0, 0, -10167391, 12477607625] [2, 2] 23068672 \(\Gamma_0(N)\)-optimal*
491970.ci3 491970ci6 [1, 0, 0, -10008691, 12886006205] [2] 46137344  
491970.ci4 491970ci4 [1, 0, 0, -1957311, -821926359] [2] 23068672  
491970.ci5 491970ci2 [1, 0, 0, -645391, 188514425] [2, 2] 11534336 \(\Gamma_0(N)\)-optimal*
491970.ci6 491970ci1 [1, 0, 0, 31729, 12327801] [2] 5767168 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 491970.ci6.

Rank

sage: E.rank()
 

The elliptic curves in class 491970.ci have rank \(0\).

Modular form 491970.2.a.ci

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

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.