Properties

Label 10766.e
Number of curves $2$
Conductor $10766$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 10766.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10766.e1 10766h1 \([1, -1, 1, -450273, 116420913]\) \(-10096027515422913423969/1328135939293184\) \(-1328135939293184\) \([7]\) \(204624\) \(1.9219\) \(\Gamma_0(N)\)-optimal
10766.e2 10766h2 \([1, -1, 1, 2885367, -4130671167]\) \(2656605299081089600290591/8905789086154159331384\) \(-8905789086154159331384\) \([]\) \(1432368\) \(2.8948\)  

Rank

sage: E.rank()
 

The elliptic curves in class 10766.e have rank \(1\).

Complex multiplication

The elliptic curves in class 10766.e do not have complex multiplication.

Modular form 10766.2.a.e

sage: E.q_eigenform(10)
 
\(q + q^{2} - 3 q^{3} + q^{4} - q^{5} - 3 q^{6} + q^{7} + q^{8} + 6 q^{9} - q^{10} + 5 q^{11} - 3 q^{12} + q^{14} + 3 q^{15} + q^{16} + 4 q^{17} + 6 q^{18} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.