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SageMath
E = EllipticCurve("cs1")
E.isogeny_class()
Elliptic curves in class 68400.cs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68400.cs1 | 68400ec3 | \([0, 0, 0, -2769600, -1774082000]\) | \(-50357871050752/19\) | \(-886464000000\) | \([]\) | \(559872\) | \(2.0806\) | |
| 68400.cs2 | 68400ec2 | \([0, 0, 0, -33600, -2522000]\) | \(-89915392/6859\) | \(-320013504000000\) | \([]\) | \(186624\) | \(1.5313\) | |
| 68400.cs3 | 68400ec1 | \([0, 0, 0, 2400, -2000]\) | \(32768/19\) | \(-886464000000\) | \([]\) | \(62208\) | \(0.98200\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68400.cs have rank \(1\).
Complex multiplication
The elliptic curves in class 68400.cs do not have complex multiplication.Modular form 68400.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
