Properties

Label 486720dv
Number of curves $6$
Conductor $486720$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("486720.dv1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 486720dv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
486720.dv6 486720dv1 [0, 0, 0, 1458132, -271408592] [2] 16515072 \(\Gamma_0(N)\)-optimal*
486720.dv5 486720dv2 [0, 0, 0, -6329388, -2255668688] [2, 2] 33030144 \(\Gamma_0(N)\)-optimal*
486720.dv3 486720dv3 [0, 0, 0, -55001388, 155422142512] [2, 2] 66060288 \(\Gamma_0(N)\)-optimal*
486720.dv2 486720dv4 [0, 0, 0, -82257708, -286926126032] [2] 66060288  
486720.dv1 486720dv5 [0, 0, 0, -877558188, 10006033356592] [2] 132120576 \(\Gamma_0(N)\)-optimal*
486720.dv4 486720dv6 [0, 0, 0, -11196588, 396190845232] [2] 132120576  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 486720dv1.

Rank

sage: E.rank()
 

The elliptic curves in class 486720dv have rank \(0\).

Modular form 486720.2.a.dv

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.