Properties

Label 205350ek
Number of curves $4$
Conductor $205350$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ek1")
 
E.isogeny_class()
 

Elliptic curves in class 205350ek

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
205350.z3 205350ek1 \([1, 1, 0, -10696025, 16548265125]\) \(-3375675045001/999000000\) \(-40049385665484375000000\) \([2]\) \(33094656\) \(3.0514\) \(\Gamma_0(N)\)-optimal
205350.z2 205350ek2 \([1, 1, 0, -181821025, 943532390125]\) \(16581570075765001/998001000\) \(40009336279818890625000\) \([2]\) \(66189312\) \(3.3980\)  
205350.z4 205350ek3 \([1, 1, 0, 79144600, -137708088000]\) \(1367594037332999/995878502400\) \(-39924246465047654400000000\) \([2]\) \(99283968\) \(3.6007\)  
205350.z1 205350ek4 \([1, 1, 0, -358935400, -1168510328000]\) \(127568139540190201/59114336463360\) \(2369862722102412701160000000\) \([2]\) \(198567936\) \(3.9473\)  

Rank

sage: E.rank()
 

The elliptic curves in class 205350ek have rank \(1\).

Complex multiplication

The elliptic curves in class 205350ek do not have complex multiplication.

Modular form 205350.2.a.ek

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} + 4 q^{7} - q^{8} + q^{9} + 6 q^{11} - q^{12} + 2 q^{13} - 4 q^{14} + q^{16} - 6 q^{17} - q^{18} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.