Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 259182.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259182.z1 259182z1 \([1, -1, 0, -395148, 95312080]\) \(142653955969251/677915392\) \(32426148683706624\) \([2]\) \(4055040\) \(2.0174\) \(\Gamma_0(N)\)-optimal
259182.z2 259182z2 \([1, -1, 0, -191868, 193089760]\) \(-16330999139811/327112335088\) \(-15646485297442473936\) \([2]\) \(8110080\) \(2.3640\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 259182.2.a.z

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2q^{5} + q^{7} - q^{8} + 2q^{10} + 4q^{13} - q^{14} + q^{16} - q^{17} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.