Properties

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

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Show commands for: SageMath
sage: E = EllipticCurve("139650.cc1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 139650jk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
139650.cc3 139650jk1 [1, 1, 0, -38000, -2850000] [2] 589824 \(\Gamma_0(N)\)-optimal
139650.cc2 139650jk2 [1, 1, 0, -62500, 1241500] [2, 2] 1179648  
139650.cc1 139650jk3 [1, 1, 0, -760750, 254706250] [2] 2359296  
139650.cc4 139650jk4 [1, 1, 0, 243750, 10122750] [2] 2359296  

Rank

sage: E.rank()
 

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

Modular form 139650.2.a.cc

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.