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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 45414o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
45414.a3 | 45414o1 | \([1, -1, 0, 1104, -10184]\) | \(9261/8\) | \(-128481837336\) | \([]\) | \(48384\) | \(0.81914\) | \(\Gamma_0(N)\)-optimal |
45414.a1 | 45414o2 | \([1, -1, 0, -24126, -1455022]\) | \(-132651/2\) | \(-23415814854486\) | \([]\) | \(145152\) | \(1.3684\) | |
45414.a2 | 45414o3 | \([1, -1, 0, -11511, 633181]\) | \(-1167051/512\) | \(-74005538305536\) | \([]\) | \(145152\) | \(1.3684\) |
Rank
sage: E.rank()
The elliptic curves in class 45414o have rank \(2\).
Complex multiplication
The elliptic curves in class 45414o do not have complex multiplication.Modular form 45414.2.a.o
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.
\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.