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Properties

Label 486720kx
Number of curves $2$
Conductor $486720$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("kx1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 486720kx have rank \(1\).

Complex multiplication

The elliptic curves in class 486720kx do not have complex multiplication.

Modular form 486720.2.a.kx

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - 2 q^{7} + 3 q^{11} - 6 q^{17} + 2 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 486720kx

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.kx1 486720kx1 \([0, 0, 0, -82598412, -289650124816]\) \(-2365581049/6750\) \(-177829884399705587712000\) \([]\) \(60383232\) \(3.3329\) \(\Gamma_0(N)\)-optimal*
486720.kx2 486720kx2 \([0, 0, 0, 164168628, -1484298718864]\) \(18573478391/46875000\) \(-1234929752775733248000000000\) \([]\) \(181149696\) \(3.8822\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 486720kx1.