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Properties

Label 416955.cs
Number of curves $2$
Conductor $416955$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("cs1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 416955.cs

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
416955.cs1 416955cs2 [0, -1, 1, -9671310, 218863581893] [] 202500000 \(\Gamma_0(N)\)-optimal*
416955.cs2 416955cs1 [0, -1, 1, -3227460, -2612452327] [] 40500000 \(\Gamma_0(N)\)-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 416955.cs1.

Rank

sage: E.rank()
 

The elliptic curves in class 416955.cs have rank \(1\).

Complex multiplication

The elliptic curves in class 416955.cs do not have complex multiplication.

Modular form 416955.2.a.cs

sage: E.q_eigenform(10)
 
\( q + 2q^{2} - q^{3} + 2q^{4} + q^{5} - 2q^{6} + q^{7} + q^{9} + 2q^{10} + q^{11} - 2q^{12} + 6q^{13} + 2q^{14} - q^{15} - 4q^{16} - 7q^{17} + 2q^{18} + O(q^{20}) \)

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.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.