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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 219450e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 219450.hn4 | 219450e1 | \([1, 0, 0, -74282978, -246429572928]\) | \(362644936777023929703844469/152957840999619348\) | \(19119730124952418500\) | \([2]\) | \(31232000\) | \(3.0437\) | \(\Gamma_0(N)\)-optimal |
| 219450.hn2 | 219450e2 | \([1, 0, 0, -1188527528, -15771198888078]\) | \(1485393209947610215717798177109/264254237478\) | \(33031779684750\) | \([2]\) | \(62464000\) | \(3.3902\) | |
| 219450.hn3 | 219450e3 | \([1, 0, 0, -341793003, 2214900059697]\) | \(35326710774513978982545998789/3488878768082999418344448\) | \(436109846010374927293056000\) | \([10]\) | \(156160000\) | \(3.8484\) | |
| 219450.hn1 | 219450e4 | \([1, 0, 0, -1239337803, -14349289224303]\) | \(1684157835362218257442567295429/263040958377251477770663008\) | \(32880119797156434721332876000\) | \([10]\) | \(312320000\) | \(4.1950\) |
Rank
sage: E.rank()
The elliptic curves in class 219450e have rank \(0\).
Complex multiplication
The elliptic curves in class 219450e do not have complex multiplication.Modular form 219450.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
