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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 4620k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4620.k2 | 4620k1 | \([0, 1, 0, 259, 4095]\) | \(7476617216/31444875\) | \(-8049888000\) | \([3]\) | \(2592\) | \(0.58332\) | \(\Gamma_0(N)\)-optimal |
4620.k1 | 4620k2 | \([0, 1, 0, -2381, -123681]\) | \(-5833703071744/22107421875\) | \(-5659500000000\) | \([]\) | \(7776\) | \(1.1326\) |
Rank
sage: E.rank()
The elliptic curves in class 4620k have rank \(1\).
Complex multiplication
The elliptic curves in class 4620k do not have complex multiplication.Modular form 4620.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.