Properties

Label 405600.bf
Number of curves $2$
Conductor $405600$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bf1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 405600.bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.bf1 405600bf2 \([0, -1, 0, -170408, -7246188]\) \(14172488/7605\) \(293663059560000000\) \([2]\) \(3096576\) \(2.0430\) \(\Gamma_0(N)\)-optimal*
405600.bf2 405600bf1 \([0, -1, 0, 40842, -908688]\) \(1560896/975\) \(-4706138775000000\) \([2]\) \(1548288\) \(1.6964\) \(\Gamma_0(N)\)-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 405600.bf1.

Rank

sage: E.rank()
 

The elliptic curves in class 405600.bf have rank \(1\).

Complex multiplication

The elliptic curves in class 405600.bf do not have complex multiplication.

Modular form 405600.2.a.bf

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2q^{7} + q^{9} - 4q^{17} + 2q^{19} + O(q^{20})\)  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.