Properties

Label 105710i
Number of curves $2$
Conductor $105710$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 105710i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
105710.f1 105710i1 \([1, 1, 1, -981, -32717]\) \(-117649/440\) \(-390501619640\) \([]\) \(119880\) \(0.90989\) \(\Gamma_0(N)\)-optimal
105710.f2 105710i2 \([1, 1, 1, 8629, 770679]\) \(80062991/332750\) \(-295316849852750\) \([]\) \(359640\) \(1.4592\)  

Rank

sage: E.rank()
 

The elliptic curves in class 105710i have rank \(0\).

Complex multiplication

The elliptic curves in class 105710i do not have complex multiplication.

Modular form 105710.2.a.i

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{5} - q^{6} - q^{7} + q^{8} - 2 q^{9} - q^{10} + q^{11} - q^{12} - 2 q^{13} - q^{14} + q^{15} + q^{16} + 3 q^{17} - 2 q^{18} - q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.