Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 259182.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259182.e1 259182e2 \([1, -1, 0, -139959, -3793091]\) \(234770924809/130960928\) \(169131843702515232\) \([2]\) \(4915200\) \(1.9961\)  
259182.e2 259182e1 \([1, -1, 0, 34281, -482531]\) \(3449795831/2071552\) \(-2675343054117888\) \([2]\) \(2457600\) \(1.6495\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 259182.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 4q^{5} - q^{7} - q^{8} + 4q^{10} + 4q^{13} + q^{14} + q^{16} - q^{17} + 6q^{19} + O(q^{20})\)  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.