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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 304920.cz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
304920.cz1 | 304920cz4 | \([0, 0, 0, -17744924307, -805818360558706]\) | \(233632133015204766393938/29145526885986328125\) | \(77087773517577539062500000000000\) | \([2]\) | \(943718400\) | \(4.8482\) | |
304920.cz2 | 304920cz2 | \([0, 0, 0, -4432422027, 100542057171446]\) | \(7282213870869695463556/912102595400390625\) | \(1206222117275004131610000000000\) | \([2, 2]\) | \(471859200\) | \(4.5016\) | |
304920.cz3 | 304920cz1 | \([0, 0, 0, -4289523447, 108132058088714]\) | \(26401417552259125806544/507547744790625\) | \(167803303714631372541600000\) | \([2]\) | \(235929600\) | \(4.1550\) | \(\Gamma_0(N)\)-optimal |
304920.cz4 | 304920cz3 | \([0, 0, 0, 6593702973, 521142416196446]\) | \(11986661998777424518222/51295853620928503125\) | \(-135673757478792867481710393600000\) | \([2]\) | \(943718400\) | \(4.8482\) |
Rank
sage: E.rank()
The elliptic curves in class 304920.cz have rank \(0\).
Complex multiplication
The elliptic curves in class 304920.cz do not have complex multiplication.Modular form 304920.2.a.cz
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.