Properties

Label 86394l
Number of curves $4$
Conductor $86394$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("l1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 86394l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86394.i4 86394l1 \([1, 1, 0, 119, -134075]\) \(103823/4386816\) \(-7771512139776\) \([2]\) \(276480\) \(1.1523\) \(\Gamma_0(N)\)-optimal
86394.i3 86394l2 \([1, 1, 0, -38601, -2883195]\) \(3590714269297/73410624\) \(130051398464064\) \([2, 2]\) \(552960\) \(1.4989\)  
86394.i2 86394l3 \([1, 1, 0, -82161, 4757229]\) \(34623662831857/14438442312\) \(25578581300689032\) \([2]\) \(1105920\) \(1.8455\)  
86394.i1 86394l4 \([1, 1, 0, -614561, -185692899]\) \(14489843500598257/6246072\) \(11065297558392\) \([2]\) \(1105920\) \(1.8455\)  

Rank

sage: E.rank()
 

The elliptic curves in class 86394l have rank \(1\).

Complex multiplication

The elliptic curves in class 86394l do not have complex multiplication.

Modular form 86394.2.a.l

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.