Properties

Label 487872ds
Number of curves $4$
Conductor $487872$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ds1")
 
E.isogeny_class()
 

Elliptic curves in class 487872ds

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
487872.ds4 487872ds1 \([0, 0, 0, 242484, -28004240]\) \(4657463/3696\) \(-1251282940965421056\) \([2]\) \(5898240\) \(2.1605\) \(\Gamma_0(N)\)-optimal*
487872.ds3 487872ds2 \([0, 0, 0, -1151436, -242667920]\) \(498677257/213444\) \(72261589840753065984\) \([2, 2]\) \(11796480\) \(2.5071\) \(\Gamma_0(N)\)-optimal*
487872.ds2 487872ds3 \([0, 0, 0, -8817996, 9910924144]\) \(223980311017/4278582\) \(1448516414535095549952\) \([2]\) \(23592960\) \(2.8536\) \(\Gamma_0(N)\)-optimal*
487872.ds1 487872ds4 \([0, 0, 0, -15787596, -24134735504]\) \(1285429208617/614922\) \(208182199303121928192\) \([2]\) \(23592960\) \(2.8536\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 487872ds1.

Rank

sage: E.rank()
 

The elliptic curves in class 487872ds have rank \(1\).

Complex multiplication

The elliptic curves in class 487872ds do not have complex multiplication.

Modular form 487872.2.a.ds

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} - q^{7} + 2 q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.