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SageMath
E = EllipticCurve("pk1")
E.isogeny_class()
Elliptic curves in class 187200.pk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.pk1 | 187200ql4 | \([0, 0, 0, -7238700, 4490046000]\) | \(520300455507/193072360\) | \(15565796400660480000000\) | \([2]\) | \(15925248\) | \(2.9582\) | |
| 187200.pk2 | 187200ql2 | \([0, 0, 0, -6398700, 6229966000]\) | \(261984288445803/42250\) | \(4672512000000000\) | \([2]\) | \(5308416\) | \(2.4089\) | |
| 187200.pk3 | 187200ql1 | \([0, 0, 0, -398700, 97966000]\) | \(-63378025803/812500\) | \(-89856000000000000\) | \([2]\) | \(2654208\) | \(2.0624\) | \(\Gamma_0(N)\)-optimal |
| 187200.pk4 | 187200ql3 | \([0, 0, 0, 1401300, 498366000]\) | \(3774555693/3515200\) | \(-283400935833600000000\) | \([2]\) | \(7962624\) | \(2.6117\) |
Rank
sage: E.rank()
The elliptic curves in class 187200.pk have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.pk do not have complex multiplication.Modular form 187200.2.a.pk
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
