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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 2166.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2166.d1 | 2166d3 | \([1, 0, 1, -31606280, 68389818806]\) | \(74220219816682217473/16416\) | \(772305182496\) | \([2]\) | \(86400\) | \(2.5737\) | |
| 2166.d2 | 2166d2 | \([1, 0, 1, -1975400, 1068459446]\) | \(18120364883707393/269485056\) | \(12678161875854336\) | \([2, 2]\) | \(43200\) | \(2.2271\) | |
| 2166.d3 | 2166d4 | \([1, 0, 1, -1917640, 1133889974]\) | \(-16576888679672833/2216253521952\) | \(-104265599459584679712\) | \([2]\) | \(86400\) | \(2.5737\) | |
| 2166.d4 | 2166d1 | \([1, 0, 1, -127080, 15656374]\) | \(4824238966273/537919488\) | \(25306896220028928\) | \([2]\) | \(21600\) | \(1.8805\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2166.d have rank \(1\).
Complex multiplication
The elliptic curves in class 2166.d do not have complex multiplication.Modular form 2166.2.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.
