Properties

Label 489762s
Number of curves $2$
Conductor $489762$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 489762s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
489762.s1 489762s1 \([1, -1, 0, -87333, 9943649]\) \(1703492778807/2384732\) \(103124279863332\) \([2]\) \(2598912\) \(1.5926\) \(\Gamma_0(N)\)-optimal
489762.s2 489762s2 \([1, -1, 0, -62763, 15638975]\) \(-632291491287/2072502446\) \(-89622365221225746\) \([2]\) \(5197824\) \(1.9392\)  

Rank

sage: E.rank()
 

The elliptic curves in class 489762s have rank \(2\).

Complex multiplication

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

Modular form 489762.2.a.s

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} + q^{7} - q^{8} + 2 q^{10} - 4 q^{11} - q^{14} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.