Properties

Label 106722.f
Number of curves $2$
Conductor $106722$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 106722.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
106722.f1 106722cj2 \([1, -1, 0, -112557693474, -11790893107312236]\) \(779828911477214942771/154308452600236032\) \(31206111149180758352193804375447552\) \([2]\) \(1570111488\) \(5.3352\)  
106722.f2 106722cj1 \([1, -1, 0, -106547110434, -13385639031234156]\) \(661452718394879874611/36407410163712\) \(7362744354426065047706016940032\) \([2]\) \(785055744\) \(4.9887\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 106722.f have rank \(1\).

Complex multiplication

The elliptic curves in class 106722.f do not have complex multiplication.

Modular form 106722.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.