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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 18480.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18480.cs1 | 18480da3 | \([0, 1, 0, -1427374880, -20757026649612]\) | \(78519570041710065450485106721/96428056919040\) | \(394969321140387840\) | \([2]\) | \(4423680\) | \(3.5506\) | |
| 18480.cs2 | 18480da5 | \([0, 1, 0, -419817760, 3030228557300]\) | \(1997773216431678333214187041/187585177195046990066400\) | \(768348885790912471311974400\) | \([4]\) | \(8847360\) | \(3.8972\) | |
| 18480.cs3 | 18480da4 | \([0, 1, 0, -93225760, -293563545100]\) | \(21876183941534093095979041/3572502915711058560000\) | \(14632971942752495861760000\) | \([2, 4]\) | \(4423680\) | \(3.5506\) | |
| 18480.cs4 | 18480da2 | \([0, 1, 0, -89211680, -324345116172]\) | \(19170300594578891358373921/671785075055001600\) | \(2751631667425286553600\) | \([2, 2]\) | \(2211840\) | \(3.2040\) | |
| 18480.cs5 | 18480da1 | \([0, 1, 0, -5325600, -5544457740]\) | \(-4078208988807294650401/880065599546327040\) | \(-3604748695741755555840\) | \([2]\) | \(1105920\) | \(2.8574\) | \(\Gamma_0(N)\)-optimal |
| 18480.cs6 | 18480da6 | \([0, 1, 0, 169140960, -1647270873612]\) | \(130650216943167617311657439/361816948816603087500000\) | \(-1482002222352806246400000000\) | \([8]\) | \(8847360\) | \(3.8972\) |
Rank
sage: E.rank()
The elliptic curves in class 18480.cs have rank \(1\).
Complex multiplication
The elliptic curves in class 18480.cs do not have complex multiplication.Modular form 18480.2.a.cs
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
