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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 3330x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3330.ba3 | 3330x1 | \([1, -1, 1, -1756247, -895392529]\) | \(821774646379511057449/38361600000\) | \(27965606400000\) | \([2]\) | \(46080\) | \(2.0589\) | \(\Gamma_0(N)\)-optimal |
| 3330.ba2 | 3330x2 | \([1, -1, 1, -1759127, -892306321]\) | \(825824067562227826729/5613755625000000\) | \(4092427850625000000\) | \([2, 2]\) | \(92160\) | \(2.4055\) | |
| 3330.ba1 | 3330x3 | \([1, -1, 1, -2884127, 388843679]\) | \(3639478711331685826729/2016912141902025000\) | \(1470328951446576225000\) | \([2]\) | \(184320\) | \(2.7521\) | |
| 3330.ba4 | 3330x4 | \([1, -1, 1, -680207, -1975973569]\) | \(-47744008200656797609/2286529541015625000\) | \(-1666880035400390625000\) | \([2]\) | \(184320\) | \(2.7521\) |
Rank
sage: E.rank()
The elliptic curves in class 3330x have rank \(0\).
Complex multiplication
The elliptic curves in class 3330x do not have complex multiplication.Modular form 3330.2.a.x
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.
