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SageMath
E = EllipticCurve("pq1")
E.isogeny_class()
Elliptic curves in class 273600.pq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 273600.pq1 | 273600pq3 | \([0, 0, 0, -43776300, 111482498000]\) | \(3107086841064961/570\) | \(1702010880000000\) | \([2]\) | \(14155776\) | \(2.7583\) | |
| 273600.pq2 | 273600pq4 | \([0, 0, 0, -3168300, 1154882000]\) | \(1177918188481/488703750\) | \(1459261578240000000000\) | \([2]\) | \(14155776\) | \(2.7583\) | |
| 273600.pq3 | 273600pq2 | \([0, 0, 0, -2736300, 1741538000]\) | \(758800078561/324900\) | \(970146201600000000\) | \([2, 2]\) | \(7077888\) | \(2.4117\) | |
| 273600.pq4 | 273600pq1 | \([0, 0, 0, -144300, 36002000]\) | \(-111284641/123120\) | \(-367634350080000000\) | \([2]\) | \(3538944\) | \(2.0651\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 273600.pq have rank \(1\).
Complex multiplication
The elliptic curves in class 273600.pq do not have complex multiplication.Modular form 273600.2.a.pq
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
