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Properties

Label 86190.b
Number of curves $4$
Conductor $86190$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 86190.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86190.b1 86190c4 \([1, 1, 0, -12948783, 17898665517]\) \(49745123032831462081/97939634471640\) \(472735909124422196760\) \([2]\) \(7741440\) \(2.8552\)  
86190.b2 86190c3 \([1, 1, 0, -10853183, -13695181443]\) \(29291056630578924481/175463302795560\) \(846927849103334168040\) \([2]\) \(7741440\) \(2.8552\)  
86190.b3 86190c2 \([1, 1, 0, -1084983, 72119637]\) \(29263955267177281/16463793153600\) \(79467584967934862400\) \([2, 2]\) \(3870720\) \(2.5086\)  
86190.b4 86190c1 \([1, 1, 0, 267017, 9116437]\) \(436192097814719/259683840000\) \(-1253444296066560000\) \([2]\) \(1935360\) \(2.1620\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 86190.b have rank \(1\).

Complex multiplication

The elliptic curves in class 86190.b do not have complex multiplication.

Modular form 86190.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{10} + 4 q^{11} - q^{12} + 4 q^{14} + q^{15} + q^{16} - q^{17} - q^{18} - 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.