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SageMath
E = EllipticCurve("po1")
E.isogeny_class()
Elliptic curves in class 187200.po
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.po1 | 187200gd4 | \([0, 0, 0, -273900, -53822000]\) | \(3044193988/85293\) | \(63670883328000000\) | \([2]\) | \(2097152\) | \(2.0034\) | |
| 187200.po2 | 187200gd2 | \([0, 0, 0, -39900, 1870000]\) | \(37642192/13689\) | \(2554695936000000\) | \([2, 2]\) | \(1048576\) | \(1.6569\) | |
| 187200.po3 | 187200gd1 | \([0, 0, 0, -35400, 2563000]\) | \(420616192/117\) | \(1364688000000\) | \([2]\) | \(524288\) | \(1.3103\) | \(\Gamma_0(N)\)-optimal |
| 187200.po4 | 187200gd3 | \([0, 0, 0, 122100, 13210000]\) | \(269676572/257049\) | \(-191886050304000000\) | \([2]\) | \(2097152\) | \(2.0034\) |
Rank
sage: E.rank()
The elliptic curves in class 187200.po have rank \(0\).
Complex multiplication
The elliptic curves in class 187200.po do not have complex multiplication.Modular form 187200.2.a.po
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.
