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SageMath
sage: E = EllipticCurve("bt1")
sage: E.isogeny_class()
Elliptic curves in class 195195bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
|---|---|---|---|---|---|
| 195195.bu6 | 195195bt1 | [1, 0, 1, 5911, -50818129] | [2] | 2211840 | \(\Gamma_0(N)\)-optimal |
| 195195.bu5 | 195195bt2 | [1, 0, 1, -2022934, -1087963693] | [2, 2] | 4423680 | |
| 195195.bu4 | 195195bt3 | [1, 0, 1, -4300209, 1809641017] | [2] | 8847360 | |
| 195195.bu2 | 195195bt4 | [1, 0, 1, -32207179, -70354769119] | [2, 2] | 8847360 | |
| 195195.bu3 | 195195bt5 | [1, 0, 1, -32047474, -71086984603] | [2] | 17694720 | |
| 195195.bu1 | 195195bt6 | [1, 0, 1, -515314804, -4502577363919] | [2] | 17694720 |
Rank
sage: E.rank()
The elliptic curves in class 195195bt have rank \(1\).
Complex multiplication
The elliptic curves in class 195195bt do not have complex multiplication.Modular form 195195.2.a.bt
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.
