Properties

Label 458640.nl
Number of curves $4$
Conductor $458640$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("nl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 458640.nl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.nl1 458640nl4 \([0, 0, 0, -3669267, -2705310286]\) \(31103978031362/195\) \(34251558082560\) \([2]\) \(7077888\) \(2.2026\)  
458640.nl2 458640nl3 \([0, 0, 0, -317667, -6778366]\) \(20183398562/11567205\) \(2031768173899376640\) \([2]\) \(7077888\) \(2.2026\) \(\Gamma_0(N)\)-optimal*
458640.nl3 458640nl2 \([0, 0, 0, -229467, -42217126]\) \(15214885924/38025\) \(3339526913049600\) \([2, 2]\) \(3538944\) \(1.8561\) \(\Gamma_0(N)\)-optimal*
458640.nl4 458640nl1 \([0, 0, 0, -8967, -1160026]\) \(-3631696/24375\) \(-535180595040000\) \([2]\) \(1769472\) \(1.5095\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 458640.nl1.

Rank

sage: E.rank()
 

The elliptic curves in class 458640.nl have rank \(0\).

Complex multiplication

The elliptic curves in class 458640.nl do not have complex multiplication.

Modular form 458640.2.a.nl

sage: E.q_eigenform(10)
 
\(q + q^{5} + 4q^{11} - q^{13} + 6q^{17} + O(q^{20})\)  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 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.