Properties

Label 409101bt
Number of curves 2
Conductor 409101
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("409101.bt1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 409101bt

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.bt2 409101bt1 [1, 0, 1, -136491, -282253919] [2] 6912000 \(\Gamma_0(N)\)-optimal*
409101.bt1 409101bt2 [1, 0, 1, -7340226, -7595485691] [2] 13824000 \(\Gamma_0(N)\)-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 409101bt1.

Rank

sage: E.rank()
 

The elliptic curves in class 409101bt have rank \(1\).

Modular form 409101.2.a.bt

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

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 Cremona 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 Cremona labels.