Properties

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

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

Elliptic curves in class 139650.cc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139650.cc1 139650jk3 \([1, 1, 0, -760750, 254706250]\) \(26487576322129/44531250\) \(81860266113281250\) \([2]\) \(2359296\) \(2.1406\)  
139650.cc2 139650jk2 \([1, 1, 0, -62500, 1241500]\) \(14688124849/8122500\) \(14931312539062500\) \([2, 2]\) \(1179648\) \(1.7940\)  
139650.cc3 139650jk1 \([1, 1, 0, -38000, -2850000]\) \(3301293169/22800\) \(41912456250000\) \([2]\) \(589824\) \(1.4474\) \(\Gamma_0(N)\)-optimal
139650.cc4 139650jk4 \([1, 1, 0, 243750, 10122750]\) \(871257511151/527800050\) \(-970236688788281250\) \([2]\) \(2359296\) \(2.1406\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 139650.2.a.cc

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.