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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 212160.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.d1 | 212160hd4 | \([0, -1, 0, -360961, -62106239]\) | \(79364416584061444/20404090514925\) | \(1337202475986124800\) | \([2]\) | \(3145728\) | \(2.1874\) | |
| 212160.d2 | 212160hd2 | \([0, -1, 0, -126961, 16658161]\) | \(13813960087661776/714574355625\) | \(11707586242560000\) | \([2, 2]\) | \(1572864\) | \(1.8409\) | |
| 212160.d3 | 212160hd1 | \([0, -1, 0, -125341, 17121805]\) | \(212670222886967296/616241925\) | \(631031731200\) | \([2]\) | \(786432\) | \(1.4943\) | \(\Gamma_0(N)\)-optimal |
| 212160.d4 | 212160hd3 | \([0, -1, 0, 81119, 65723425]\) | \(900753985478876/29018422265625\) | \(-1901751321600000000\) | \([2]\) | \(3145728\) | \(2.1874\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.d have rank \(2\).
Complex multiplication
The elliptic curves in class 212160.d do not have complex multiplication.Modular form 212160.2.a.d
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 LMFDB 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 LMFDB labels.
