Properties

Label 429b
Number of curves 6
Conductor 429
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("429.b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 429b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
429.b5 429b1 [1, 0, 0, -24, 63] [4] 64 \(\Gamma_0(N)\)-optimal
429.b4 429b2 [1, 0, 0, -429, 3384] [2, 4] 128  
429.b3 429b3 [1, 0, 0, -474, 2619] [2, 2] 256  
429.b1 429b4 [1, 0, 0, -6864, 218313] [4] 256  
429.b2 429b5 [1, 0, 0, -3009, -61770] [2] 512  
429.b6 429b6 [1, 0, 0, 1341, 18228] [2] 512  

Rank

sage: E.rank()
 

The elliptic curves in class 429b have rank \(1\).

Modular form 429.2.a.b

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} + 3q^{8} + q^{9} + 2q^{10} - q^{11} - q^{12} + q^{13} - 2q^{15} - q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.