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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 5200.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5200.x1 | 5200p3 | \([0, 1, 0, -183808, 30270388]\) | \(-10730978619193/6656\) | \(-425984000000\) | \([]\) | \(15552\) | \(1.5523\) | |
5200.x2 | 5200p2 | \([0, 1, 0, -1808, 58388]\) | \(-10218313/17576\) | \(-1124864000000\) | \([]\) | \(5184\) | \(1.0029\) | |
5200.x3 | 5200p1 | \([0, 1, 0, 192, -1612]\) | \(12167/26\) | \(-1664000000\) | \([]\) | \(1728\) | \(0.45364\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5200.x have rank \(0\).
Complex multiplication
The elliptic curves in class 5200.x do not have complex multiplication.Modular form 5200.2.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 LMFDB numbering.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.