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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)
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.