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Properties

Label 458640.ee
Number of curves $4$
Conductor $458640$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ee1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 458640.ee have rank \(1\).

Complex multiplication

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

Modular form 458640.2.a.ee

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} + q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 458640.ee

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
458640.ee1 458640ee4 \([0, 0, 0, -180839324883, -29599579022441582]\) \(1861772567578966373029167169/9401133413380800000\) \(3302599663080082275847372800000\) \([2]\) \(1698693120\) \(5.0541\)  
458640.ee2 458640ee2 \([0, 0, 0, -11495324883, -445891752041582]\) \(478202393398338853167169/32244226560000000000\) \(11327333321508346920960000000000\) \([2, 2]\) \(849346560\) \(4.7076\)  
458640.ee3 458640ee1 \([0, 0, 0, -2246884563, 32568909161362]\) \(3571003510905229697089/762141946675200000\) \(267738965678984195683123200000\) \([2]\) \(424673280\) \(4.3610\) \(\Gamma_0(N)\)-optimal*
458640.ee4 458640ee3 \([0, 0, 0, 9873629997, -1913686798629998]\) \(303025056761573589385151/4678857421875000000000\) \(-1643673402516600000000000000000000\) \([2]\) \(1698693120\) \(5.0541\)  
*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.ee1.