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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 48510bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
48510.y3 | 48510bf1 | \([1, -1, 0, -437040, -76538624]\) | \(107639597521009/32699842560\) | \(2804538653681909760\) | \([2]\) | \(983040\) | \(2.2448\) | \(\Gamma_0(N)\)-optimal |
48510.y2 | 48510bf2 | \([1, -1, 0, -2694960, 1644448000]\) | \(25238585142450289/995844326400\) | \(85409704995185894400\) | \([2, 2]\) | \(1966080\) | \(2.5914\) | |
48510.y4 | 48510bf3 | \([1, -1, 0, 1185840, 5988615520]\) | \(2150235484224911/181905111732960\) | \(-15601295823407567048160\) | \([2]\) | \(3932160\) | \(2.9380\) | |
48510.y1 | 48510bf4 | \([1, -1, 0, -42702480, 107416329376]\) | \(100407751863770656369/166028940000\) | \(14239658157541740000\) | \([2]\) | \(3932160\) | \(2.9380\) |
Rank
sage: E.rank()
The elliptic curves in class 48510bf have rank \(0\).
Complex multiplication
The elliptic curves in class 48510bf do not have complex multiplication.Modular form 48510.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.