Properties

Label 137280ez
Number of curves $4$
Conductor $137280$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ez1")
 
E.isogeny_class()
 

Elliptic curves in class 137280ez

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
137280.dx4 137280ez1 \([0, 1, 0, 4159, -132705]\) \(30342134159/47190000\) \(-12370575360000\) \([2]\) \(393216\) \(1.1975\) \(\Gamma_0(N)\)-optimal
137280.dx3 137280ez2 \([0, 1, 0, -27841, -1367905]\) \(9104453457841/2226896100\) \(583767451238400\) \([2, 2]\) \(786432\) \(1.5440\)  
137280.dx2 137280ez3 \([0, 1, 0, -152641, 21770015]\) \(1500376464746641/83599963590\) \(21915228855336960\) \([2]\) \(1572864\) \(1.8906\)  
137280.dx1 137280ez4 \([0, 1, 0, -415041, -103046625]\) \(30161840495801041/2799263610\) \(733810159779840\) \([2]\) \(1572864\) \(1.8906\)  

Rank

sage: E.rank()
 

The elliptic curves in class 137280ez have rank \(1\).

Complex multiplication

The elliptic curves in class 137280ez do not have complex multiplication.

Modular form 137280.2.a.ez

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - 4 q^{7} + q^{9} + q^{11} - q^{13} - q^{15} + 6 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.