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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 177870cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.gb4 | 177870cs1 | \([1, 1, 1, 1520665, 961579037]\) | \(1865864036231/2993760000\) | \(-623966584615244640000\) | \([4]\) | \(7372800\) | \(2.6754\) | \(\Gamma_0(N)\)-optimal |
| 177870.gb3 | 177870cs2 | \([1, 1, 1, -10337335, 9798160637]\) | \(586145095611769/140040608400\) | \(29187596911839606147600\) | \([2, 2]\) | \(14745600\) | \(3.0220\) | |
| 177870.gb1 | 177870cs3 | \([1, 1, 1, -154412035, 738412733477]\) | \(1953542217204454969/170843779260\) | \(35607667096768935154140\) | \([2]\) | \(29491200\) | \(3.3685\) | |
| 177870.gb2 | 177870cs4 | \([1, 1, 1, -55990635, -153038029803]\) | \(93137706732176569/5369647977540\) | \(1119154811718972015201060\) | \([2]\) | \(29491200\) | \(3.3685\) |
Rank
sage: E.rank()
The elliptic curves in class 177870cs have rank \(1\).
Complex multiplication
The elliptic curves in class 177870cs do not have complex multiplication.Modular form 177870.2.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 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.
