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SageMath
E = EllipticCurve("pt1")
E.isogeny_class()
Elliptic curves in class 187200.pt
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
187200.pt1 | 187200cc2 | \([0, 0, 0, -223500, -13250000]\) | \(26463592/13689\) | \(638673984000000000\) | \([2]\) | \(2293760\) | \(2.1085\) | |
187200.pt2 | 187200cc1 | \([0, 0, 0, -178500, -29000000]\) | \(107850176/117\) | \(682344000000000\) | \([2]\) | \(1146880\) | \(1.7620\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.pt have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.pt do not have complex multiplication.Modular form 187200.2.a.pt
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.