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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 71632.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
71632.f1 | 71632p3 | \([0, -1, 0, -3626773, -2657242531]\) | \(727057727488000/37\) | \(268483612672\) | \([]\) | \(583200\) | \(2.1142\) | |
71632.f2 | 71632p2 | \([0, -1, 0, -45173, -3563459]\) | \(1404928000/50653\) | \(367554065747968\) | \([]\) | \(194400\) | \(1.5649\) | |
71632.f3 | 71632p1 | \([0, -1, 0, -6453, 200125]\) | \(4096000/37\) | \(268483612672\) | \([]\) | \(64800\) | \(1.0156\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 71632.f have rank \(0\).
Complex multiplication
The elliptic curves in class 71632.f do not have complex multiplication.Modular form 71632.2.a.f
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.