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Properties

Label 429590.bw
Number of curves $2$
Conductor $429590$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bw1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 429590.bw have rank \(1\).

Complex multiplication

The elliptic curves in class 429590.bw do not have complex multiplication.

Modular form 429590.2.a.bw

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - 3 q^{9} - q^{10} + 2 q^{11} + q^{14} + q^{16} + q^{17} - 3 q^{18} + O(q^{20})\) 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 429590.bw

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
429590.bw1 429590bw2 \([1, -1, 1, -1496413, 702840517]\) \(7876916680687209/27200448800\) \(1279669077391392800\) \([2]\) \(6289920\) \(2.3375\) \(\Gamma_0(N)\)-optimal*
429590.bw2 429590bw1 \([1, -1, 1, -52413, 20694917]\) \(-338463151209/3731840000\) \(-175567700551040000\) \([2]\) \(3144960\) \(1.9909\) \(\Gamma_0(N)\)-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 429590.bw1.