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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 369600.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
369600.j1 | 369600j4 | \([0, -1, 0, -154929633, -742196692863]\) | \(100407751863770656369/166028940000\) | \(680054538240000000000\) | \([2]\) | \(47185920\) | \(3.2601\) | |
369600.j2 | 369600j2 | \([0, -1, 0, -9777633, -11356372863]\) | \(25238585142450289/995844326400\) | \(4078978360934400000000\) | \([2, 2]\) | \(23592960\) | \(2.9136\) | |
369600.j3 | 369600j1 | \([0, -1, 0, -1585633, 530219137]\) | \(107639597521009/32699842560\) | \(133938555125760000000\) | \([2]\) | \(11796480\) | \(2.5670\) | \(\Gamma_0(N)\)-optimal |
369600.j4 | 369600j3 | \([0, -1, 0, 4302367, -41389012863]\) | \(2150235484224911/181905111732960\) | \(-745083337658204160000000\) | \([2]\) | \(47185920\) | \(3.2601\) |
Rank
sage: E.rank()
The elliptic curves in class 369600.j have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.j do not have complex multiplication.Modular form 369600.2.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.