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Properties

Label 413270.bu
Number of curves $2$
Conductor $413270$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bu1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 413270.bu have rank \(0\).

Complex multiplication

The elliptic curves in class 413270.bu do not have complex multiplication.

Modular form 413270.2.a.bu

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - 2 q^{3} + q^{4} - q^{5} - 2 q^{6} - 4 q^{7} + q^{8} + q^{9} - q^{10} + q^{11} - 2 q^{12} + q^{13} - 4 q^{14} + 2 q^{15} + q^{16} + q^{18} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 413270.bu

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
413270.bu1 413270bu2 \([1, 0, 0, -31796, -1422410]\) \(147281603041/49156250\) \(1186512376156250\) \([2]\) \(2949120\) \(1.5951\) \(\Gamma_0(N)\)-optimal*
413270.bu2 413270bu1 \([1, 0, 0, 5774, -152544]\) \(881974079/929500\) \(-22435870385500\) \([2]\) \(1474560\) \(1.2485\) \(\Gamma_0(N)\)-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 413270.bu1.