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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 574d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
574.a2 | 574d1 | \([1, 0, 1, -31679, 5254674]\) | \(-3515753329334380009/9905620513718272\) | \(-9905620513718272\) | \([2]\) | \(8160\) | \(1.7565\) | \(\Gamma_0(N)\)-optimal |
574.a1 | 574d2 | \([1, 0, 1, -687039, 218902034]\) | \(35864681248144538691049/43574618474283008\) | \(43574618474283008\) | \([2]\) | \(16320\) | \(2.1031\) |
Rank
sage: E.rank()
The elliptic curves in class 574d have rank \(0\).
Complex multiplication
The elliptic curves in class 574d do not have complex multiplication.Modular form 574.2.a.d
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.