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Properties

Label 5586r
Number of curves $2$
Conductor $5586$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 5586r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5586.p1 5586r1 \([1, 0, 1, -4681, 122840]\) \(96386901625/18468\) \(2172741732\) \([2]\) \(5760\) \(0.79226\) \(\Gamma_0(N)\)-optimal
5586.p2 5586r2 \([1, 0, 1, -4191, 149692]\) \(-69173457625/42633378\) \(-5015774288322\) \([2]\) \(11520\) \(1.1388\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5586r have rank \(1\).

Complex multiplication

The elliptic curves in class 5586r do not have complex multiplication.

Modular form 5586.2.a.r

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.