Properties

Label 422331bq
Number of curves $6$
Conductor $422331$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("422331.bq1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 422331bq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
422331.bq4 422331bq1 [1, 0, 1, -4463632, -3630149911] [2] 8257536 \(\Gamma_0(N)\)-optimal*
422331.bq3 422331bq2 [1, 0, 1, -4505037, -3559380485] [2, 2] 16515072 \(\Gamma_0(N)\)-optimal*
422331.bq2 422331bq3 [1, 0, 1, -11502482, 10183601495] [2, 2] 33030144 \(\Gamma_0(N)\)-optimal*
422331.bq5 422331bq4 [1, 0, 1, 1829928, -12772953581] [2] 33030144  
422331.bq1 422331bq5 [1, 0, 1, -167061067, 830972919389] [2] 66060288 \(\Gamma_0(N)\)-optimal*
422331.bq6 422331bq6 [1, 0, 1, 32096983, 68973120101] [2] 66060288  
*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 422331bq1.

Rank

sage: E.rank()
 

The elliptic curves in class 422331bq have rank \(1\).

Modular form 422331.2.a.bq

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

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.