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Properties

Label 165886k
Number of curves $2$
Conductor $165886$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 165886k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
165886.be2 165886k1 \([1, -1, 1, -5592927, 4687786567]\) \(801581275315909089/70810888830976\) \(1709202715109012537344\) \([]\) \(17781120\) \(2.8144\) \(\Gamma_0(N)\)-optimal
165886.be1 165886k2 \([1, -1, 1, -2777669367, -56346064121513]\) \(98191033604529537629349729/10906239337336\) \(263250104535461976184\) \([]\) \(124467840\) \(3.7873\)  

Rank

sage: E.rank()
 

The elliptic curves in class 165886k have rank \(1\).

Complex multiplication

The elliptic curves in class 165886k do not have complex multiplication.

Modular form 165886.2.a.k

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.