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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 134640cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 134640.bb4 | 134640cc1 | \([0, 0, 0, -510843, -23053142]\) | \(4937402992298041/2780405760000\) | \(8302247112867840000\) | \([2]\) | \(1769472\) | \(2.3203\) | \(\Gamma_0(N)\)-optimal |
| 134640.bb2 | 134640cc2 | \([0, 0, 0, -5118843, 4434726058]\) | \(4967657717692586041/29490113030400\) | \(88057005666965913600\) | \([2, 2]\) | \(3538944\) | \(2.6669\) | |
| 134640.bb1 | 134640cc3 | \([0, 0, 0, -81784443, 284678160298]\) | \(20260414982443110947641/720358602480\) | \(2150979261267640320\) | \([2]\) | \(7077888\) | \(3.0135\) | |
| 134640.bb3 | 134640cc4 | \([0, 0, 0, -2181243, 9489160618]\) | \(-384369029857072441/12804787777021680\) | \(-38234891425582304133120\) | \([2]\) | \(7077888\) | \(3.0135\) |
Rank
sage: E.rank()
The elliptic curves in class 134640cc have rank \(0\).
Complex multiplication
The elliptic curves in class 134640cc do not have complex multiplication.Modular form 134640.2.a.cc
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.
