Properties

Label 479808.ni
Number of curves $6$
Conductor $479808$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ni1")
 
E.isogeny_class()
 

Elliptic curves in class 479808.ni

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
479808.ni1 479808ni5 \([0, 0, 0, -387193879884, 92734389270274192]\) \(285531136548675601769470657/17941034271597192\) \(403369601585106552706100625408\) \([2]\) \(2264924160\) \(5.1411\) \(\Gamma_0(N)\)-optimal*
479808.ni2 479808ni3 \([0, 0, 0, -24245658444, 1443184565779600]\) \(70108386184777836280897/552468975892674624\) \(12421200880640252920624186392576\) \([2, 2]\) \(1132462080\) \(4.7945\) \(\Gamma_0(N)\)-optimal*
479808.ni3 479808ni6 \([0, 0, 0, -8258455884, 3317952650942608]\) \(-2770540998624539614657/209924951154647363208\) \(-4719758216173604166756801227784192\) \([2]\) \(2264924160\) \(5.1411\)  
479808.ni4 479808ni2 \([0, 0, 0, -2560594764, -12533759058800]\) \(82582985847542515777/44772582831427584\) \(1006625294018868876956575727616\) \([2, 2]\) \(566231040\) \(4.4480\) \(\Gamma_0(N)\)-optimal*
479808.ni5 479808ni1 \([0, 0, 0, -1982567244, -33934881170288]\) \(38331145780597164097/55468445663232\) \(1247101169810717006383546368\) \([2]\) \(283115520\) \(4.1014\) \(\Gamma_0(N)\)-optimal*
479808.ni6 479808ni4 \([0, 0, 0, 9876028596, -98580268761968]\) \(4738217997934888496063/2928751705237796928\) \(-65847341385090655872686827241472\) \([2]\) \(1132462080\) \(4.7945\)  
*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 479808.ni1.

Rank

sage: E.rank()
 

The elliptic curves in class 479808.ni have rank \(1\).

Complex multiplication

The elliptic curves in class 479808.ni do not have complex multiplication.

Modular form 479808.2.a.ni

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.