Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(8619\)\(\medspace = 3 \cdot 13^{2} \cdot 17 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.1.8619.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.51.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.8619.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - x^{2} + 9x + 12 \)
|
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 17\cdot 19 + 5\cdot 19^{2} + 12\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 17\cdot 19 + 10\cdot 19^{2} + 12\cdot 19^{3} + 15\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 3\cdot 19 + 2\cdot 19^{2} + 13\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)
|
Generators of the action on the roots $ r_{ 1 }, r_{ 2 }, r_{ 3 } $
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $ r_{ 1 }, r_{ 2 }, r_{ 3 } $ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $3$ | $2$ | $(1,2)$ | $0$ | ✓ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |