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SageMath
E = EllipticCurve("ke1")
E.isogeny_class()
Elliptic curves in class 381150ke
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 381150.ke2 | 381150ke1 | \([1, -1, 1, 121945, 13721447]\) | \(397535/392\) | \(-197756032753125000\) | \([]\) | \(4860000\) | \(2.0058\) | \(\Gamma_0(N)\)-optimal |
| 381150.ke1 | 381150ke2 | \([1, -1, 1, -1239305, -745856053]\) | \(-417267265/235298\) | \(-118703058660063281250\) | \([]\) | \(14580000\) | \(2.5551\) |
Rank
sage: E.rank()
The elliptic curves in class 381150ke have rank \(1\).
Complex multiplication
The elliptic curves in class 381150ke do not have complex multiplication.Modular form 381150.2.a.ke
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
