Properties

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

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Show commands: SageMath
E = EllipticCurve("bx1")
 
E.isogeny_class()
 

Elliptic curves in class 409101.bx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
409101.bx1 409101bx2 \([1, 0, 1, -55371055, -158523172909]\) \(262623524091319/134454573\) \(9612006346648997569971\) \([2]\) \(38707200\) \(3.1697\) \(\Gamma_0(N)\)-optimal*
409101.bx2 409101bx1 \([1, 0, 1, -2869760, -3350345407]\) \(-36561310759/46662561\) \(-3335854054461174739647\) \([2]\) \(19353600\) \(2.8231\) \(\Gamma_0(N)\)-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.bx1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 409101.2.a.bx

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