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Properties

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

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

Elliptic curves in class 422331bq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422331.bq4 422331bq1 \([1, 0, 1, -4463632, -3630149911]\) \(17319700013617/25857\) \(14683395250024137\) \([2]\) \(8257536\) \(2.3709\) \(\Gamma_0(N)\)-optimal*
422331.bq3 422331bq2 \([1, 0, 1, -4505037, -3559380485]\) \(17806161424897/668584449\) \(379668550979874110409\) \([2, 2]\) \(16515072\) \(2.7174\) \(\Gamma_0(N)\)-optimal*
422331.bq2 422331bq3 \([1, 0, 1, -11502482, 10183601495]\) \(296380748763217/92608836489\) \(52589710709395698524049\) \([2, 2]\) \(33030144\) \(3.0640\) \(\Gamma_0(N)\)-optimal*
422331.bq5 422331bq4 \([1, 0, 1, 1829928, -12772953581]\) \(1193377118543/124806800313\) \(-70873944343373754688833\) \([2]\) \(33030144\) \(3.0640\)  
422331.bq1 422331bq5 \([1, 0, 1, -167061067, 830972919389]\) \(908031902324522977/161726530797\) \(91839524078878141606677\) \([2]\) \(66060288\) \(3.4106\) \(\Gamma_0(N)\)-optimal*
422331.bq6 422331bq6 \([1, 0, 1, 32096983, 68973120101]\) \(6439735268725823/7345472585373\) \(-4171268022943436026996293\) \([2]\) \(66060288\) \(3.4106\)  
*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 422331bq1.

Rank

sage: E.rank()
 

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

Complex multiplication

The elliptic curves in class 422331bq do not have complex multiplication.

Modular form 422331.2.a.bq

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