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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 133200.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 133200.q1 | 133200bs4 | \([0, 0, 0, -37755075, -39858002750]\) | \(127568139540190201/59114336463360\) | \(2758038482034524160000000\) | \([2]\) | \(27869184\) | \(3.3843\) | |
| 133200.q2 | 133200bs2 | \([0, 0, 0, -19125075, 32190687250]\) | \(16581570075765001/998001000\) | \(46562734656000000000\) | \([2]\) | \(9289728\) | \(2.8350\) | |
| 133200.q3 | 133200bs1 | \([0, 0, 0, -1125075, 564687250]\) | \(-3375675045001/999000000\) | \(-46609344000000000000\) | \([2]\) | \(4644864\) | \(2.4884\) | \(\Gamma_0(N)\)-optimal |
| 133200.q4 | 133200bs3 | \([0, 0, 0, 8324925, -4698962750]\) | \(1367594037332999/995878502400\) | \(-46463707407974400000000\) | \([2]\) | \(13934592\) | \(3.0377\) |
Rank
sage: E.rank()
The elliptic curves in class 133200.q have rank \(1\).
Complex multiplication
The elliptic curves in class 133200.q do not have complex multiplication.Modular form 133200.2.a.q
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
