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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 5712.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5712.f1 | 5712g3 | \([0, -1, 0, -181367424, -940067661600]\) | \(322159999717985454060440834/4250799\) | \(8705636352\) | \([2]\) | \(276480\) | \(2.8910\) | |
5712.f2 | 5712g4 | \([0, -1, 0, -11364624, -14606337312]\) | \(79260902459030376659234/842751810121431609\) | \(1725955707128691935232\) | \([4]\) | \(276480\) | \(2.8910\) | |
5712.f3 | 5712g2 | \([0, -1, 0, -11335464, -14685722496]\) | \(157304700372188331121828/18069292138401\) | \(18502955149722624\) | \([2, 2]\) | \(138240\) | \(2.5444\) | |
5712.f4 | 5712g1 | \([0, -1, 0, -706644, -230527296]\) | \(-152435594466395827792/1646846627220711\) | \(-421592736568502016\) | \([2]\) | \(69120\) | \(2.1979\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5712.f have rank \(1\).
Complex multiplication
The elliptic curves in class 5712.f do not have complex multiplication.Modular form 5712.2.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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.