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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 89232k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 89232.i3 | 89232k1 | \([0, -1, 0, -129679, -13133810]\) | \(3122884507648/835956693\) | \(64560052630122192\) | \([2]\) | \(903168\) | \(1.9338\) | \(\Gamma_0(N)\)-optimal |
| 89232.i2 | 89232k2 | \([0, -1, 0, -745684, 237457024]\) | \(37109806448848/1803785841\) | \(2228871611241310464\) | \([2, 2]\) | \(1806336\) | \(2.2804\) | |
| 89232.i4 | 89232k3 | \([0, -1, 0, 440696, 920811904]\) | \(1915049403068/75239967231\) | \(-371885005814062980096\) | \([4]\) | \(3612672\) | \(2.6270\) | |
| 89232.i1 | 89232k4 | \([0, -1, 0, -11788144, 15582059440]\) | \(36652193922790372/93308787\) | \(461192901499579392\) | \([2]\) | \(3612672\) | \(2.6270\) |
Rank
sage: E.rank()
The elliptic curves in class 89232k have rank \(0\).
Complex multiplication
The elliptic curves in class 89232k do not have complex multiplication.Modular form 89232.2.a.k
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.
