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SageMath
E = EllipticCurve("fs1")
E.isogeny_class()
Elliptic curves in class 145200fs
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 145200.dj6 | 145200fs1 | \([0, -1, 0, 12340992, 1581760512]\) | \(1833318007919/1070530560\) | \(-121376652121866240000000\) | \([2]\) | \(13271040\) | \(3.1189\) | \(\Gamma_0(N)\)-optimal |
| 145200.dj5 | 145200fs2 | \([0, -1, 0, -49611008, 12733120512]\) | \(119102750067601/68309049600\) | \(7744873485979238400000000\) | \([2, 2]\) | \(26542080\) | \(3.4654\) | |
| 145200.dj2 | 145200fs3 | \([0, -1, 0, -572331008, 5258751040512]\) | \(182864522286982801/463015182960\) | \(52496616994547235840000000\) | \([2]\) | \(53084160\) | \(3.8120\) | |
| 145200.dj3 | 145200fs4 | \([0, -1, 0, -518123008, -4520588991488]\) | \(135670761487282321/643043610000\) | \(72908222769613440000000000\) | \([2, 2]\) | \(53084160\) | \(3.8120\) | |
| 145200.dj4 | 145200fs5 | \([0, -1, 0, -251923008, -9159922591488]\) | \(-15595206456730321/310672490129100\) | \(-35224017106278305606400000000\) | \([2]\) | \(106168320\) | \(4.1586\) | |
| 145200.dj1 | 145200fs6 | \([0, -1, 0, -8280515008, -290021366751488]\) | \(553808571467029327441/12529687500\) | \(1420614765900000000000000\) | \([2]\) | \(106168320\) | \(4.1586\) |
Rank
sage: E.rank()
The elliptic curves in class 145200fs have rank \(1\).
Complex multiplication
The elliptic curves in class 145200fs do not have complex multiplication.Modular form 145200.2.a.fs
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
