Properties

Label 489762.dv
Number of curves $4$
Conductor $489762$
CM no
Rank $0$
Graph

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

Elliptic curves in class 489762.dv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
489762.dv1 489762dv3 \([1, -1, 1, -272029361, 1726986279971]\) \(632678989847546725777/80515134\) \(283312125428578974\) \([2]\) \(70778880\) \(3.2088\) \(\Gamma_0(N)\)-optimal*
489762.dv2 489762dv4 \([1, -1, 1, -19452101, 18704540027]\) \(231331938231569617/90942310746882\) \(320002688551514287703202\) \([2]\) \(70778880\) \(3.2088\)  
489762.dv3 489762dv2 \([1, -1, 1, -17003291, 26982497351]\) \(154502321244119857/55101928644\) \(193889567635142190084\) \([2, 2]\) \(35389440\) \(2.8622\) \(\Gamma_0(N)\)-optimal*
489762.dv4 489762dv1 \([1, -1, 1, -911111, 546264047]\) \(-23771111713777/22848457968\) \(-80397868923370737648\) \([2]\) \(17694720\) \(2.5156\) \(\Gamma_0(N)\)-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 489762.dv1.

Rank

sage: E.rank()
 

The elliptic curves in class 489762.dv have rank \(0\).

Complex multiplication

The elliptic curves in class 489762.dv do not have complex multiplication.

Modular form 489762.2.a.dv

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - 2 q^{5} + q^{7} + q^{8} - 2 q^{10} + 4 q^{11} + q^{14} + q^{16} - 6 q^{17} + O(q^{20})\) 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.