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Properties

Label 139650ge
Number of curves $2$
Conductor $139650$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ge1")
 
E.isogeny_class()
 

Elliptic curves in class 139650ge

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139650.ct2 139650ge1 \([1, 0, 1, -1783626, 920920648]\) \(-341370886042369/1817528220\) \(-3341099649293437500\) \([2]\) \(4838400\) \(2.3992\) \(\Gamma_0(N)\)-optimal
139650.ct1 139650ge2 \([1, 0, 1, -28574376, 58788940648]\) \(1403607530712116449/39475350\) \(72566178939843750\) \([2]\) \(9676800\) \(2.7457\)  

Rank

sage: E.rank()
 

The elliptic curves in class 139650ge have rank \(1\).

Complex multiplication

The elliptic curves in class 139650ge do not have complex multiplication.

Modular form 139650.2.a.ge

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} - 4 q^{11} + q^{12} + 6 q^{13} + q^{16} + 4 q^{17} - q^{18} - q^{19} + O(q^{20})\) 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.