Properties

Label 409101.bm
Number of curves $6$
Conductor $409101$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("409101.bm1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 409101.bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.bm1 409101bm6 [1, 1, 0, -552013739, -4992211246392] [2] 47185920  
409101.bm2 409101bm4 [1, 1, 0, -34500974, -78013532505] [2, 2] 23592960  
409101.bm3 409101bm5 [1, 1, 0, -33522689, -82644538038] [2] 47185920  
409101.bm4 409101bm3 [1, 1, 0, -8354084, 8056799997] [2] 23592960 \(\Gamma_0(N)\)-optimal*
409101.bm5 409101bm2 [1, 1, 0, -2217569, -1146745200] [2, 2] 11796480 \(\Gamma_0(N)\)-optimal*
409101.bm6 409101bm1 [1, 1, 0, 183676, -88756653] [2] 5898240 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 409101.bm6.

Rank

sage: E.rank()
 

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

Modular form 409101.2.a.bm

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

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.