Properties

Label 493680.ex
Number of curves $4$
Conductor $493680$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ex1")
 
E.isogeny_class()
 

Elliptic curves in class 493680.ex

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
493680.ex1 493680ex3 \([0, 1, 0, -23171056, 42922866644]\) \(189602977175292169/1402500\) \(10176980183040000\) \([2]\) \(23592960\) \(2.6666\) \(\Gamma_0(N)\)-optimal*
493680.ex2 493680ex4 \([0, 1, 0, -2029936, 81850580]\) \(127483771761289/73369857660\) \(532394714751005736960\) \([2]\) \(23592960\) \(2.6666\)  
493680.ex3 493680ex2 \([0, 1, 0, -1449136, 669387860]\) \(46380496070089/125888400\) \(913485741229670400\) \([2, 2]\) \(11796480\) \(2.3200\) \(\Gamma_0(N)\)-optimal*
493680.ex4 493680ex1 \([0, 1, 0, -55216, 18706004]\) \(-2565726409/19388160\) \(-140686574050344960\) \([2]\) \(5898240\) \(1.9734\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 493680.ex1.

Rank

sage: E.rank()
 

The elliptic curves in class 493680.ex have rank \(1\).

Complex multiplication

The elliptic curves in class 493680.ex do not have complex multiplication.

Modular form 493680.2.a.ex

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{9} + 6 q^{13} - q^{15} + q^{17} + 8 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.