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Properties

Label 490245.p
Number of curves $2$
Conductor $490245$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 490245.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
490245.p1 490245p2 \([1, 0, 0, -67621, -6772060]\) \(290656902035521/86293125\) \(10152299863125\) \([2]\) \(1695744\) \(1.4742\) \(\Gamma_0(N)\)-optimal*
490245.p2 490245p1 \([1, 0, 0, -3676, -134569]\) \(-46694890801/39169575\) \(-4608261329175\) \([2]\) \(847872\) \(1.1277\) \(\Gamma_0(N)\)-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 490245.p1.

Rank

sage: E.rank()
 

The elliptic curves in class 490245.p have rank \(0\).

Complex multiplication

The elliptic curves in class 490245.p do not have complex multiplication.

Modular form 490245.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 3 q^{8} + q^{9} + q^{10} + 2 q^{11} - q^{12} - 2 q^{13} - q^{15} - q^{16} - 4 q^{17} - q^{18} + 6 q^{19} + O(q^{20})\) 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 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.