Properties

Label 486720p
Number of curves $4$
Conductor $486720$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 486720p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.p4 486720p1 \([0, 0, 0, 1052532, 324417808]\) \(3774555693/3515200\) \(-120091934173062758400\) \([2]\) \(18579456\) \(2.5401\) \(\Gamma_0(N)\)-optimal*
486720.p3 486720p2 \([0, 0, 0, -5437068, 2922853648]\) \(520300455507/193072360\) \(6596049484455472005120\) \([2]\) \(37158912\) \(2.8867\) \(\Gamma_0(N)\)-optimal*
486720.p2 486720p3 \([0, 0, 0, -24256908, 46489961232]\) \(-63378025803/812500\) \(-20235535320858624000000\) \([2]\) \(55738368\) \(3.0894\) \(\Gamma_0(N)\)-optimal*
486720.p1 486720p4 \([0, 0, 0, -389296908, 2956442825232]\) \(261984288445803/42250\) \(1052247836684648448000\) \([2]\) \(111476736\) \(3.4360\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 486720p1.

Rank

sage: E.rank()
 

The elliptic curves in class 486720p have rank \(2\).

Complex multiplication

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

Modular form 486720.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{7} - 6 q^{17} + 4 q^{19} + O(q^{20})\) Copy content 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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.