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Properties

Label 491970.co
Number of curves $4$
Conductor $491970$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("co1")
 
E.isogeny_class()
 

Elliptic curves in class 491970.co

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
491970.co1 491970co4 \([1, 0, 0, -336164170, 2372302687400]\) \(28379906689597370652529/1357352437500\) \(200936874771629437500\) \([2]\) \(109486080\) \(3.3731\) \(\Gamma_0(N)\)-optimal*
491970.co2 491970co3 \([1, 0, 0, -20975390, 37195091892]\) \(-6894246873502147249/47925198774000\) \(-7094649406010800086000\) \([2]\) \(54743040\) \(3.0265\) \(\Gamma_0(N)\)-optimal*
491970.co3 491970co2 \([1, 0, 0, -4512910, 2651349572]\) \(68663623745397169/19216056254400\) \(2844665970694114161600\) \([2]\) \(36495360\) \(2.8238\) \(\Gamma_0(N)\)-optimal*
491970.co4 491970co1 \([1, 0, 0, 734770, 272051460]\) \(296354077829711/387386634240\) \(-57347124786436239360\) \([2]\) \(18247680\) \(2.4772\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 491970.co1.

Rank

sage: E.rank()
 

The elliptic curves in class 491970.co have rank \(2\).

Complex multiplication

The elliptic curves in class 491970.co do not have complex multiplication.

Modular form 491970.2.a.co

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{10} + q^{12} - 4 q^{13} - 2 q^{14} + q^{15} + q^{16} - 6 q^{17} + q^{18} - 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.