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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 338800cw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
338800.cw2 | 338800cw1 | \([0, -1, 0, 216792, 32524912]\) | \(397535/392\) | \(-1111123059200000000\) | \([]\) | \(3888000\) | \(2.1496\) | \(\Gamma_0(N)\)-optimal |
338800.cw1 | 338800cw2 | \([0, -1, 0, -2203208, -1767955088]\) | \(-417267265/235298\) | \(-666951616284800000000\) | \([]\) | \(11664000\) | \(2.6989\) |
Rank
sage: E.rank()
The elliptic curves in class 338800cw have rank \(1\).
Complex multiplication
The elliptic curves in class 338800cw do not have complex multiplication.Modular form 338800.2.a.cw
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.