Properties

Label 139650.do
Number of curves $4$
Conductor $139650$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("do1")
 
E.isogeny_class()
 

Elliptic curves in class 139650.do

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139650.do1 139650gs3 \([1, 0, 1, -524326, 146088548]\) \(8671983378625/82308\) \(151303967062500\) \([2]\) \(1492992\) \(1.8832\)  
139650.do2 139650gs4 \([1, 0, 1, -512076, 153242548]\) \(-8078253774625/846825858\) \(-1556690865122531250\) \([2]\) \(2985984\) \(2.2297\)  
139650.do3 139650gs1 \([1, 0, 1, -9826, -29452]\) \(57066625/32832\) \(60353937000000\) \([2]\) \(497664\) \(1.3339\) \(\Gamma_0(N)\)-optimal
139650.do4 139650gs2 \([1, 0, 1, 39174, -225452]\) \(3616805375/2105352\) \(-3870196210125000\) \([2]\) \(995328\) \(1.6804\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 139650.2.a.do

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.