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SageMath
E = EllipticCurve("gd1")
E.isogeny_class()
Elliptic curves in class 76050gd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76050.fw2 | 76050gd1 | \([1, -1, 1, -491315, 2263187]\) | \(29819839301/17252352\) | \(7588325745321984000\) | \([2]\) | \(2408448\) | \(2.3122\) | \(\Gamma_0(N)\)-optimal |
76050.fw1 | 76050gd2 | \([1, -1, 1, -5358515, -4757858413]\) | \(38686490446661/141927552\) | \(62425836014250384000\) | \([2]\) | \(4816896\) | \(2.6588\) |
Rank
sage: E.rank()
The elliptic curves in class 76050gd have rank \(0\).
Complex multiplication
The elliptic curves in class 76050gd do not have complex multiplication.Modular form 76050.2.a.gd
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.