Properties

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

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Show commands: SageMath
E = EllipticCurve("he1")
 
E.isogeny_class()
 

Elliptic curves in class 106722.he

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
106722.he1 106722gy2 \([1, -1, 1, -57127874, 166207834433]\) \(46546832455691959/748268928\) \(331463315979518760576\) \([2]\) \(10321920\) \(3.0704\)  
106722.he2 106722gy1 \([1, -1, 1, -3461954, 2762908481]\) \(-10358806345399/1445216256\) \(-640192522495421693952\) \([2]\) \(5160960\) \(2.7238\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 106722.2.a.he

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