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SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 141570ef
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
141570.z2 | 141570ef1 | \([1, -1, 0, -47341575, 179222376361]\) | \(-9085904860560159241/5484993611139900\) | \(-7083693558956822427863100\) | \([2]\) | \(30965760\) | \(3.4705\) | \(\Gamma_0(N)\)-optimal |
141570.z1 | 141570ef2 | \([1, -1, 0, -844544025, 9445425333691]\) | \(51583042491609575206441/9586057511268810\) | \(12380086224795524663746890\) | \([2]\) | \(61931520\) | \(3.8171\) |
Rank
sage: E.rank()
The elliptic curves in class 141570ef have rank \(0\).
Complex multiplication
The elliptic curves in class 141570ef do not have complex multiplication.Modular form 141570.2.a.ef
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.