Properties

Label 424830.x
Number of curves $4$
Conductor $424830$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("x1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 424830.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
424830.x1 424830x4 \([1, 1, 0, -5289428, -4684521582]\) \(5763259856089/5670\) \(16101444049443270\) \([2]\) \(14155776\) \(2.4031\)  
424830.x2 424830x2 \([1, 1, 0, -333078, -72142272]\) \(1439069689/44100\) \(125233453717892100\) \([2, 2]\) \(7077888\) \(2.0566\)  
424830.x3 424830x1 \([1, 1, 0, -49858, 2684452]\) \(4826809/1680\) \(4770798236872080\) \([2]\) \(3538944\) \(1.7100\) \(\Gamma_0(N)\)-optimal*
424830.x4 424830x3 \([1, 1, 0, 91752, -243008898]\) \(30080231/9003750\) \(-25568496800736303750\) \([2]\) \(14155776\) \(2.4031\)  
*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 424830.x1.

Rank

sage: E.rank()
 

The elliptic curves in class 424830.x have rank \(0\).

Complex multiplication

The elliptic curves in class 424830.x do not have complex multiplication.

Modular form 424830.2.a.x

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} + 2q^{13} + q^{15} + q^{16} - q^{18} + O(q^{20})\)  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 LMFDB 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 LMFDB labels.