Properties

Label 490049.c
Number of curves $2$
Conductor $490049$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("490049.c1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 490049.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
490049.c1 490049c2 [1, -1, 0, -2644000, -1613913477] [2] 13436928 \(\Gamma_0(N)\)-optimal*
490049.c2 490049c1 [1, -1, 0, -2626115, -1637360712] [2] 6718464 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 490049.c2.

Rank

sage: E.rank()
 

The elliptic curves in class 490049.c have rank \(1\).

Modular form 490049.2.a.c

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{4} + 4q^{5} - 3q^{8} - 3q^{9} + 4q^{10} + 4q^{11} - q^{16} - 2q^{17} - 3q^{18} + 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.