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Properties

Label 490110.w
Number of curves $1$
Conductor $490110$
CM no
Rank $2$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 1, -51112840119, 4458855244956826]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -51112840119, 4458855244956826]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -51112840119, 4458855244956826]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curve 490110.w1 has rank \(2\).

Complex multiplication

The elliptic curves in class 490110.w do not have complex multiplication.

Modular form 490110.2.a.w

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - 5 q^{7} - q^{8} + q^{9} + q^{10} + q^{11} + q^{12} + 2 q^{13} + 5 q^{14} - q^{15} + q^{16} - q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

Elliptic curves in class 490110.w

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
490110.w1 490110w1 \([1, 0, 1, -51112840119, 4458855244956826]\) \(-17314908540888569072648809/49990800256096665600\) \(-42636705492928093608764856729600\) \([]\) \(3059804160\) \(4.9403\) \(\Gamma_0(N)\)-optimal