Properties

Label 259182x
Number of curves $2$
Conductor $259182$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 259182x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259182.x2 259182x1 \([1, -1, 0, -49803, 2630645]\) \(521465741649/188737808\) \(4944566871843984\) \([2]\) \(1548288\) \(1.7120\) \(\Gamma_0(N)\)-optimal
259182.x1 259182x2 \([1, -1, 0, -340863, -74616679]\) \(167183982669969/4730963524\) \(123942127762089252\) \([2]\) \(3096576\) \(2.0586\)  

Rank

sage: E.rank()
 

The elliptic curves in class 259182x have rank \(1\).

Complex multiplication

The elliptic curves in class 259182x do not have complex multiplication.

Modular form 259182.2.a.x

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} + q^{7} - q^{8} + 2 q^{10} - q^{14} + q^{16} + q^{17} - 2 q^{19} + O(q^{20})\) 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 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.