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SageMath
E = EllipticCurve("es1")
E.isogeny_class()
Elliptic curves in class 33600es
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33600.f3 | 33600es1 | \([0, -1, 0, -7133, 224637]\) | \(2508888064/118125\) | \(1890000000000\) | \([2]\) | \(73728\) | \(1.1164\) | \(\Gamma_0(N)\)-optimal |
| 33600.f2 | 33600es2 | \([0, -1, 0, -19633, -762863]\) | \(3269383504/893025\) | \(228614400000000\) | \([2, 2]\) | \(147456\) | \(1.4630\) | |
| 33600.f4 | 33600es3 | \([0, -1, 0, 50367, -5032863]\) | \(13799183324/18600435\) | \(-19046845440000000\) | \([2]\) | \(294912\) | \(1.8096\) | |
| 33600.f1 | 33600es4 | \([0, -1, 0, -289633, -59892863]\) | \(2624033547076/324135\) | \(331914240000000\) | \([2]\) | \(294912\) | \(1.8096\) |
Rank
sage: E.rank()
The elliptic curves in class 33600es have rank \(2\).
Complex multiplication
The elliptic curves in class 33600es do not have complex multiplication.Modular form 33600.2.a.es
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
