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SageMath
E = EllipticCurve("ds1")
E.isogeny_class()
Elliptic curves in class 34650ds
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.el3 | 34650ds1 | \([1, -1, 1, -222980, 27988647]\) | \(107639597521009/32699842560\) | \(372471644160000000\) | \([2]\) | \(491520\) | \(2.0766\) | \(\Gamma_0(N)\)-optimal |
| 34650.el2 | 34650ds2 | \([1, -1, 1, -1374980, -598699353]\) | \(25238585142450289/995844326400\) | \(11343289280400000000\) | \([2, 2]\) | \(983040\) | \(2.4231\) | |
| 34650.el4 | 34650ds3 | \([1, -1, 1, 605020, -2182699353]\) | \(2150235484224911/181905111732960\) | \(-2072012913333247500000\) | \([2]\) | \(1966080\) | \(2.7697\) | |
| 34650.el1 | 34650ds4 | \([1, -1, 1, -21786980, -39136555353]\) | \(100407751863770656369/166028940000\) | \(1891173394687500000\) | \([2]\) | \(1966080\) | \(2.7697\) |
Rank
sage: E.rank()
The elliptic curves in class 34650ds have rank \(1\).
Complex multiplication
The elliptic curves in class 34650ds do not have complex multiplication.Modular form 34650.2.a.ds
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 Cremona 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 Cremona labels.
