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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 48576x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
48576.cs2 | 48576x1 | \([0, 1, 0, -373, -2821]\) | \(5619712000/184437\) | \(188863488\) | \([2]\) | \(18432\) | \(0.36142\) | \(\Gamma_0(N)\)-optimal |
48576.cs1 | 48576x2 | \([0, 1, 0, -913, 6575]\) | \(5142706000/1728243\) | \(28315533312\) | \([2]\) | \(36864\) | \(0.70799\) |
Rank
sage: E.rank()
The elliptic curves in class 48576x have rank \(2\).
Complex multiplication
The elliptic curves in class 48576x do not have complex multiplication.Modular form 48576.2.a.x
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.