Properties

Label 486720.eu
Number of curves $6$
Conductor $486720$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("486720.eu1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 486720.eu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
486720.eu1 486720eu6 [0, 0, 0, -877558188, -10006033356592] [2] 132120576  
486720.eu2 486720eu3 [0, 0, 0, -82257708, 286926126032] [2] 66060288 \(\Gamma_0(N)\)-optimal*
486720.eu3 486720eu4 [0, 0, 0, -55001388, -155422142512] [2, 2] 66060288  
486720.eu4 486720eu5 [0, 0, 0, -11196588, -396190845232] [2] 132120576  
486720.eu5 486720eu2 [0, 0, 0, -6329388, 2255668688] [2, 2] 33030144 \(\Gamma_0(N)\)-optimal*
486720.eu6 486720eu1 [0, 0, 0, 1458132, 271408592] [2] 16515072 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 486720.eu6.

Rank

sage: E.rank()
 

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

Modular form 486720.2.a.eu

sage: E.q_eigenform(10)
 
\( q - q^{5} + 4q^{11} + 6q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.