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Properties

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

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bs1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

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

Complex multiplication

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

Modular form 406560.2.a.bs

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

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 406560bs

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
406560.bs3 406560bs1 \([0, -1, 0, -8510, 300192]\) \(601211584/11025\) \(1250013441600\) \([2, 2]\) \(983040\) \(1.1160\) \(\Gamma_0(N)\)-optimal*
406560.bs1 406560bs2 \([0, -1, 0, -135560, 19256052]\) \(303735479048/105\) \(95239119360\) \([2]\) \(1966080\) \(1.4625\) \(\Gamma_0(N)\)-optimal*
406560.bs4 406560bs3 \([0, -1, 0, -40, 862600]\) \(-8/354375\) \(-321432027840000\) \([2]\) \(1966080\) \(1.4625\)  
406560.bs2 406560bs4 \([0, -1, 0, -17585, -442143]\) \(82881856/36015\) \(261336143523840\) \([2]\) \(1966080\) \(1.4625\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 406560bs1.