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Properties

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

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

Elliptic curves in class 429429.y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
429429.y1 429429y5 \([1, 1, 1, -92409457, 341879946728]\) \(10206027697760497/5557167\) \(47519260433422560783\) \([2]\) \(36864000\) \(3.1042\) \(\Gamma_0(N)\)-optimal*
429429.y2 429429y3 \([1, 1, 1, -5807942, 5277178226]\) \(2533811507137/58110129\) \(496898933174184061521\) \([2, 2]\) \(18432000\) \(2.7576\) \(\Gamma_0(N)\)-optimal*
429429.y3 429429y2 \([1, 1, 1, -797937, -153667194]\) \(6570725617/2614689\) \(22358170546864112961\) \([2, 2]\) \(9216000\) \(2.4111\) \(\Gamma_0(N)\)-optimal*
429429.y4 429429y1 \([1, 1, 1, -695692, -223561876]\) \(4354703137/1617\) \(13826945297998833\) \([2]\) \(4608000\) \(2.0645\) \(\Gamma_0(N)\)-optimal*
429429.y5 429429y6 \([1, 1, 1, 633493, 16348716704]\) \(3288008303/13504609503\) \(-115477734612749664202047\) \([2]\) \(36864000\) \(3.1042\)  
429429.y6 429429y4 \([1, 1, 1, 2576148, -1109208066]\) \(221115865823/190238433\) \(-1626726287364264703617\) \([2]\) \(18432000\) \(2.7576\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 429429.y1.

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 429429.y do not have complex multiplication.

Modular form 429429.2.a.y

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