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Properties

Label 490050h
Number of curves $2$
Conductor $490050$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("h1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 490050h have rank \(0\).

Complex multiplication

The elliptic curves in class 490050h do not have complex multiplication.

Modular form 490050.2.a.h

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 4 q^{7} - q^{8} - 4 q^{13} + 4 q^{14} + q^{16} - 3 q^{17} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 490050h

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
490050.h1 490050h1 \([1, -1, 0, -6012, -177974]\) \(-6699465/2\) \(-7174822050\) \([]\) \(729000\) \(0.86942\) \(\Gamma_0(N)\)-optimal*
490050.h2 490050h2 \([1, -1, 0, 3063, -657739]\) \(135/8\) \(-188296029880200\) \([]\) \(2187000\) \(1.4187\) \(\Gamma_0(N)\)-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 490050h1.