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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 25872.be
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
25872.be1 | 25872bk4 | \([0, -1, 0, -114872, 15009072]\) | \(347873904937/395307\) | \(190494610403328\) | \([2]\) | \(110592\) | \(1.6535\) | |
25872.be2 | 25872bk2 | \([0, -1, 0, -9032, 106800]\) | \(169112377/88209\) | \(42507061825536\) | \([2, 2]\) | \(55296\) | \(1.3069\) | |
25872.be3 | 25872bk1 | \([0, -1, 0, -5112, -137808]\) | \(30664297/297\) | \(143121420288\) | \([2]\) | \(27648\) | \(0.96032\) | \(\Gamma_0(N)\)-optimal |
25872.be4 | 25872bk3 | \([0, -1, 0, 34088, 796720]\) | \(9090072503/5845851\) | \(-2817058915528704\) | \([2]\) | \(110592\) | \(1.6535\) |
Rank
sage: E.rank()
The elliptic curves in class 25872.be have rank \(1\).
Complex multiplication
The elliptic curves in class 25872.be do not have complex multiplication.Modular form 25872.2.a.be
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.