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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 81733c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
81733.c3 | 81733c1 | \([0, 1, 1, -7363, -243747]\) | \(4096000/37\) | \(398830967173\) | \([]\) | \(70380\) | \(1.0485\) | \(\Gamma_0(N)\)-optimal |
81733.c2 | 81733c2 | \([0, 1, 1, -51543, 4344346]\) | \(1404928000/50653\) | \(545999594059837\) | \([]\) | \(211140\) | \(1.5978\) | |
81733.c1 | 81733c3 | \([0, 1, 1, -4138193, 3238764355]\) | \(727057727488000/37\) | \(398830967173\) | \([]\) | \(633420\) | \(2.1472\) |
Rank
sage: E.rank()
The elliptic curves in class 81733c have rank \(1\).
Complex multiplication
The elliptic curves in class 81733c do not have complex multiplication.Modular form 81733.2.a.c
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.