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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 217800.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
217800.j1 | 217800bc4 | \([0, 0, 0, -2913075, -1913645250]\) | \(132304644/5\) | \(103317437520000000\) | \([2]\) | \(3932160\) | \(2.3508\) | |
217800.j2 | 217800bc2 | \([0, 0, 0, -190575, -26952750]\) | \(148176/25\) | \(129146796900000000\) | \([2, 2]\) | \(1966080\) | \(2.0042\) | |
217800.j3 | 217800bc1 | \([0, 0, 0, -54450, 4492125]\) | \(55296/5\) | \(1614334961250000\) | \([2]\) | \(983040\) | \(1.6576\) | \(\Gamma_0(N)\)-optimal |
217800.j4 | 217800bc3 | \([0, 0, 0, 353925, -152732250]\) | \(237276/625\) | \(-12914679690000000000\) | \([2]\) | \(3932160\) | \(2.3508\) |
Rank
sage: E.rank()
The elliptic curves in class 217800.j have rank \(2\).
Complex multiplication
The elliptic curves in class 217800.j do not have complex multiplication.Modular form 217800.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.