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SageMath
E = EllipticCurve("pe1")
E.isogeny_class()
Elliptic curves in class 187200.pe
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.pe1 | 187200bw2 | \([0, 0, 0, -8940, 106000]\) | \(26463592/13689\) | \(40875134976000\) | \([2]\) | \(458752\) | \(1.3038\) | |
| 187200.pe2 | 187200bw1 | \([0, 0, 0, -7140, 232000]\) | \(107850176/117\) | \(43670016000\) | \([2]\) | \(229376\) | \(0.95725\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 187200.pe have rank \(1\).
Complex multiplication
The elliptic curves in class 187200.pe do not have complex multiplication.Modular form 187200.2.a.pe
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.
