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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1287.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1287.e1 | 1287e3 | \([1, -1, 0, -61776, -5894451]\) | \(35765103905346817/1287\) | \(938223\) | \([2]\) | \(2048\) | \(1.0904\) | |
| 1287.e2 | 1287e5 | \([1, -1, 0, -27081, 1667790]\) | \(3013001140430737/108679952667\) | \(79227685494243\) | \([4]\) | \(4096\) | \(1.4370\) | |
| 1287.e3 | 1287e4 | \([1, -1, 0, -4266, -70713]\) | \(11779205551777/3763454409\) | \(2743558264161\) | \([2, 2]\) | \(2048\) | \(1.0904\) | |
| 1287.e4 | 1287e2 | \([1, -1, 0, -3861, -91368]\) | \(8732907467857/1656369\) | \(1207493001\) | \([2, 2]\) | \(1024\) | \(0.74383\) | |
| 1287.e5 | 1287e1 | \([1, -1, 0, -216, -1701]\) | \(-1532808577/938223\) | \(-683964567\) | \([2]\) | \(512\) | \(0.39725\) | \(\Gamma_0(N)\)-optimal |
| 1287.e6 | 1287e6 | \([1, -1, 0, 12069, -492156]\) | \(266679605718863/296110251723\) | \(-215864373506067\) | \([2]\) | \(4096\) | \(1.4370\) |
Rank
sage: E.rank()
The elliptic curves in class 1287.e have rank \(0\).
Complex multiplication
The elliptic curves in class 1287.e do not have complex multiplication.Modular form 1287.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
