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SageMath
E = EllipticCurve("fz1")
E.isogeny_class()
Elliptic curves in class 212160fz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.fo4 | 212160fz1 | \([0, 1, 0, 59839, 4452639]\) | \(90391899763439/84690294000\) | \(-22201052430336000\) | \([2]\) | \(1622016\) | \(1.8238\) | \(\Gamma_0(N)\)-optimal |
| 212160.fo3 | 212160fz2 | \([0, 1, 0, -310081, 39890975]\) | \(12577973014374481/4642947562500\) | \(1217120845824000000\) | \([2, 2]\) | \(3244032\) | \(2.1703\) | |
| 212160.fo1 | 212160fz3 | \([0, 1, 0, -4390081, 3538082975]\) | \(35694515311673154481/10400566692750\) | \(2726446155104256000\) | \([2]\) | \(6488064\) | \(2.5169\) | |
| 212160.fo2 | 212160fz4 | \([0, 1, 0, -2148801, -1184328801]\) | \(4185743240664514801/113629394531250\) | \(29787264000000000000\) | \([2]\) | \(6488064\) | \(2.5169\) |
Rank
sage: E.rank()
The elliptic curves in class 212160fz have rank \(0\).
Complex multiplication
The elliptic curves in class 212160fz do not have complex multiplication.Modular form 212160.2.a.fz
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.
