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Properties

Label 409101.s
Number of curves $2$
Conductor $409101$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 409101.s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
409101.s1 409101s2 \([1, 0, 0, -7930161, -8593877412]\) \(264621653112601/81336717\) \(16952392145765427813\) \([2]\) \(25067520\) \(2.6665\) \(\Gamma_0(N)\)-optimal*
409101.s2 409101s1 \([1, 0, 0, -429976, -171169657]\) \(-42180533641/36293103\) \(-7564294908075226167\) \([2]\) \(12533760\) \(2.3200\) \(\Gamma_0(N)\)-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 409101.s1.

Rank

sage: E.rank()
 

The elliptic curves in class 409101.s have rank \(0\).

Complex multiplication

The elliptic curves in class 409101.s do not have complex multiplication.

Modular form 409101.2.a.s

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