Properties

Label 409101.bh
Number of curves $2$
Conductor $409101$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("409101.bh1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 409101.bh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
409101.bh1 409101bh2 [1, 1, 0, -1130021, 461682390] [2] 5529600 \(\Gamma_0(N)\)-optimal*
409101.bh2 409101bh1 [1, 1, 0, -58566, 9742671] [2] 2764800 \(\Gamma_0(N)\)-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 409101.bh2.

Rank

sage: E.rank()
 

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

Modular form 409101.2.a.bh

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

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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.