Properties

Label 429429.y
Number of curves 6
Conductor 429429
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("429429.y1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 429429.y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
429429.y1 429429y5 [1, 1, 1, -92409457, 341879946728] [u'2'] 36864000 \(\Gamma_0(N)\)-optimal*
429429.y2 429429y3 [1, 1, 1, -5807942, 5277178226] [u'2', u'2'] 18432000 \(\Gamma_0(N)\)-optimal*
429429.y3 429429y2 [1, 1, 1, -797937, -153667194] [u'2', u'2'] 9216000 \(\Gamma_0(N)\)-optimal*
429429.y4 429429y1 [1, 1, 1, -695692, -223561876] [u'2'] 4608000 \(\Gamma_0(N)\)-optimal*
429429.y5 429429y6 [1, 1, 1, 633493, 16348716704] [u'2'] 36864000  
429429.y6 429429y4 [1, 1, 1, 2576148, -1109208066] [u'2'] 18432000  
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 429429.y4.

Rank

sage: E.rank()
 

The elliptic curves in class 429429.y have rank \(1\).

Modular form 429429.2.a.y

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + q^{7} + 3q^{8} + q^{9} - 2q^{10} + q^{12} - q^{14} - 2q^{15} - q^{16} - 2q^{17} - q^{18} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.