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Properties

Label 490110.bc
Number of curves $2$
Conductor $490110$
CM no
Rank $1$
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, -44391974, -113836904584]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -44391974, -113836904584]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -44391974, -113836904584]) 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 curves in class 490110.bc have rank \(1\).

Complex multiplication

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

Modular form 490110.2.a.bc

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);
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + 2 q^{7} - q^{8} + q^{9} + q^{10} + 4 q^{11} + q^{12} - 4 q^{13} - 2 q^{14} - q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 490110.bc

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.bc1 490110bc2 \([1, 0, 1, -44391974, -113836904584]\) \(10901014250685308569/1040774054400\) \(923690804369294246400\) \([2]\) \(60963840\) \(3.0596\) \(\Gamma_0(N)\)-optimal*
490110.bc2 490110bc1 \([1, 0, 1, -2569254, -2053138568]\) \(-2113364608155289/828431400960\) \(-735235917807986933760\) \([2]\) \(30481920\) \(2.7130\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 490110.bc1.