Properties

Label 406560.fg
Number of curves $4$
Conductor $406560$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("fg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 406560.fg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
406560.fg1 406560fg4 \([0, 1, 0, -135560, -19256052]\) \(303735479048/105\) \(95239119360\) \([2]\) \(1966080\) \(1.4625\)  
406560.fg2 406560fg2 \([0, 1, 0, -17585, 442143]\) \(82881856/36015\) \(261336143523840\) \([2]\) \(1966080\) \(1.4625\) \(\Gamma_0(N)\)-optimal*
406560.fg3 406560fg1 \([0, 1, 0, -8510, -300192]\) \(601211584/11025\) \(1250013441600\) \([2, 2]\) \(983040\) \(1.1160\) \(\Gamma_0(N)\)-optimal*
406560.fg4 406560fg3 \([0, 1, 0, -40, -862600]\) \(-8/354375\) \(-321432027840000\) \([2]\) \(1966080\) \(1.4625\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 406560.fg1.

Rank

sage: E.rank()
 

The elliptic curves in class 406560.fg have rank \(1\).

Complex multiplication

The elliptic curves in class 406560.fg do not have complex multiplication.

Modular form 406560.2.a.fg

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{7} + q^{9} - 6q^{13} + q^{15} + 6q^{17} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.