Properties

Label 408980bf
Number of curves $2$
Conductor $408980$
CM no
Rank $0$
Graph

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

Elliptic curves in class 408980bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
408980.bf2 408980bf1 \([0, 1, 0, 620, 41380]\) \(176/5\) \(-747576177920\) \([]\) \(331776\) \(0.95842\) \(\Gamma_0(N)\)-optimal*
408980.bf1 408980bf2 \([0, 1, 0, -73740, 7685588]\) \(-296587984/125\) \(-18689404448000\) \([]\) \(995328\) \(1.5077\) \(\Gamma_0(N)\)-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 408980bf1.

Rank

sage: E.rank()
 

The elliptic curves in class 408980bf have rank \(0\).

Complex multiplication

The elliptic curves in class 408980bf do not have complex multiplication.

Modular form 408980.2.a.bf

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} - q^{7} - 2 q^{9} + q^{15} + 2 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.