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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 4998.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4998.k1 | 4998l2 | \([1, 0, 1, -671130, -209942708]\) | \(5799070911693913/54760833024\) | \(315685304977588224\) | \([]\) | \(90720\) | \(2.1786\) | |
4998.k2 | 4998l1 | \([1, 0, 1, -58875, 5326150]\) | \(3914907891433/135834624\) | \(783059576269824\) | \([3]\) | \(30240\) | \(1.6293\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4998.k have rank \(1\).
Complex multiplication
The elliptic curves in class 4998.k do not have complex multiplication.Modular form 4998.2.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 LMFDB 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 LMFDB labels.