Properties

Label 422331.bj
Number of curves $2$
Conductor $422331$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 422331.bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
422331.bj1 422331bj1 \([1, 1, 0, -3768027, -2815342200]\) \(10418796526321/6390657\) \(3629057610640580937\) \([2]\) \(18063360\) \(2.5041\) \(\Gamma_0(N)\)-optimal
422331.bj2 422331bj2 \([1, 1, 0, -3064142, -3898621215]\) \(-5602762882081/8312741073\) \(-4720550055535009779993\) \([2]\) \(36126720\) \(2.8507\)  

Rank

sage: E.rank()
 

The elliptic curves in class 422331.bj have rank \(0\).

Complex multiplication

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

Modular form 422331.2.a.bj

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.