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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1650a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1650.b5 | 1650a1 | \([1, 1, 0, 125, 2125]\) | \(13651919/126720\) | \(-1980000000\) | \([2]\) | \(768\) | \(0.46509\) | \(\Gamma_0(N)\)-optimal |
| 1650.b4 | 1650a2 | \([1, 1, 0, -1875, 28125]\) | \(46694890801/3920400\) | \(61256250000\) | \([2, 2]\) | \(1536\) | \(0.81166\) | |
| 1650.b3 | 1650a3 | \([1, 1, 0, -6375, -165375]\) | \(1834216913521/329422500\) | \(5147226562500\) | \([2, 2]\) | \(3072\) | \(1.1582\) | |
| 1650.b2 | 1650a4 | \([1, 1, 0, -29375, 1925625]\) | \(179415687049201/1443420\) | \(22553437500\) | \([2]\) | \(3072\) | \(1.1582\) | |
| 1650.b1 | 1650a5 | \([1, 1, 0, -97125, -11690625]\) | \(6484907238722641/283593750\) | \(4431152343750\) | \([2]\) | \(6144\) | \(1.5048\) | |
| 1650.b6 | 1650a6 | \([1, 1, 0, 12375, -934125]\) | \(13411719834479/32153832150\) | \(-502403627343750\) | \([2]\) | \(6144\) | \(1.5048\) |
Rank
sage: E.rank()
The elliptic curves in class 1650a have rank \(1\).
Complex multiplication
The elliptic curves in class 1650a do not have complex multiplication.Modular form 1650.2.a.a
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
