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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 199920cw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 199920.dt3 | 199920cw1 | \([0, -1, 0, -79200, 7517952]\) | \(114013572049/15667200\) | \(7549871770828800\) | \([2]\) | \(1327104\) | \(1.7735\) | \(\Gamma_0(N)\)-optimal |
| 199920.dt2 | 199920cw2 | \([0, -1, 0, -330080, -65337600]\) | \(8253429989329/936360000\) | \(451222805053440000\) | \([2, 2]\) | \(2654208\) | \(2.1201\) | |
| 199920.dt4 | 199920cw3 | \([0, -1, 0, 453920, -329388800]\) | \(21464092074671/109596256200\) | \(-52813373217479884800\) | \([2]\) | \(5308416\) | \(2.4666\) | |
| 199920.dt1 | 199920cw4 | \([0, -1, 0, -5128160, -4468055808]\) | \(30949975477232209/478125000\) | \(230403801600000000\) | \([2]\) | \(5308416\) | \(2.4666\) |
Rank
sage: E.rank()
The elliptic curves in class 199920cw have rank \(0\).
Complex multiplication
The elliptic curves in class 199920cw do not have complex multiplication.Modular form 199920.2.a.cw
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.
