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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 57330do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 57330.dr4 | 57330do1 | \([1, -1, 1, -102983, 12729147]\) | \(1408317602329/2153060\) | \(184659604480260\) | \([2]\) | \(331776\) | \(1.6371\) | \(\Gamma_0(N)\)-optimal |
| 57330.dr3 | 57330do2 | \([1, -1, 1, -133853, 4493031]\) | \(3092354182009/1689383150\) | \(144891839658261150\) | \([2]\) | \(663552\) | \(1.9837\) | |
| 57330.dr2 | 57330do3 | \([1, -1, 1, -418298, -91622919]\) | \(94376601570889/12235496000\) | \(1049391030431016000\) | \([2]\) | \(995328\) | \(2.1864\) | |
| 57330.dr1 | 57330do4 | \([1, -1, 1, -6468818, -6330919143]\) | \(349046010201856969/7245875000\) | \(621450592000875000\) | \([2]\) | \(1990656\) | \(2.5330\) |
Rank
sage: E.rank()
The elliptic curves in class 57330do have rank \(0\).
Complex multiplication
The elliptic curves in class 57330do do not have complex multiplication.Modular form 57330.2.a.do
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.
