Properties

Label 406560bx
Number of curves $4$
Conductor $406560$
CM no
Rank $0$
Graph

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

Elliptic curves in class 406560bx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
406560.bx3 406560bx1 \([0, -1, 0, -107730, -13145328]\) \(1219555693504/43758225\) \(4961303349710400\) \([2, 2]\) \(2211840\) \(1.7819\) \(\Gamma_0(N)\)-optimal*
406560.bx2 406560bx2 \([0, -1, 0, -271080, 36382392]\) \(2428799546888/778248135\) \(705901590675832320\) \([2]\) \(4423680\) \(2.1285\) \(\Gamma_0(N)\)-optimal*
406560.bx4 406560bx3 \([0, -1, 0, 40495, -46673823]\) \(1012048064/130203045\) \(-944794159526891520\) \([2]\) \(4423680\) \(2.1285\)  
406560.bx1 406560bx4 \([0, -1, 0, -1708560, -859023900]\) \(608119035935048/826875\) \(750008064960000\) \([2]\) \(4423680\) \(2.1285\)  
*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 406560bx1.

Rank

sage: E.rank()
 

The elliptic curves in class 406560bx have rank \(0\).

Complex multiplication

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

Modular form 406560.2.a.bx

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} - q^{7} + q^{9} - 2 q^{13} - q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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

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

The vertices are labelled with Cremona labels.