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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 490k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
490.e1 | 490k1 | \([1, -1, 1, -132, -549]\) | \(-5154200289/20\) | \(-980\) | \([]\) | \(120\) | \(-0.21282\) | \(\Gamma_0(N)\)-optimal |
490.e2 | 490k2 | \([1, -1, 1, 918, 5289]\) | \(1747829720511/1280000000\) | \(-62720000000\) | \([7]\) | \(840\) | \(0.76013\) |
Rank
sage: E.rank()
The elliptic curves in class 490k have rank \(0\).
Complex multiplication
The elliptic curves in class 490k do not have complex multiplication.Modular form 490.2.a.k
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.
\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.