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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 89232f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
89232.p2 | 89232f1 | \([0, -1, 0, 481932, 469027584]\) | \(10017976862000/82759712607\) | \(-102263123366113552128\) | \([2]\) | \(1806336\) | \(2.5209\) | \(\Gamma_0(N)\)-optimal |
89232.p1 | 89232f2 | \([0, -1, 0, -6910128, 6435898416]\) | \(7382814913718500/654774260283\) | \(3236321579522382793728\) | \([2]\) | \(3612672\) | \(2.8675\) |
Rank
sage: E.rank()
The elliptic curves in class 89232f have rank \(0\).
Complex multiplication
The elliptic curves in class 89232f do not have complex multiplication.Modular form 89232.2.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 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.