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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 17661.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17661.f1 | 17661a5 | \([1, 1, 0, -659361, -206353626]\) | \(53297461115137/147\) | \(87439028187\) | \([2]\) | \(100352\) | \(1.7579\) | |
| 17661.f2 | 17661a3 | \([1, 1, 0, -41226, -3234465]\) | \(13027640977/21609\) | \(12853537143489\) | \([2, 2]\) | \(50176\) | \(1.4113\) | |
| 17661.f3 | 17661a4 | \([1, 1, 0, -32816, 2260629]\) | \(6570725617/45927\) | \(27318450663567\) | \([2]\) | \(50176\) | \(1.4113\) | |
| 17661.f4 | 17661a6 | \([1, 1, 0, -28611, -5235204]\) | \(-4354703137/17294403\) | \(-10287114227172363\) | \([2]\) | \(100352\) | \(1.7579\) | |
| 17661.f5 | 17661a2 | \([1, 1, 0, -3381, -17640]\) | \(7189057/3969\) | \(2360853761049\) | \([2, 2]\) | \(25088\) | \(1.0647\) | |
| 17661.f6 | 17661a1 | \([1, 1, 0, 824, -1661]\) | \(103823/63\) | \(-37473869223\) | \([2]\) | \(12544\) | \(0.71813\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 17661.f have rank \(1\).
Complex multiplication
The elliptic curves in class 17661.f do not have complex multiplication.Modular form 17661.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
