Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(76\)\(\medspace = 2^{2} \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.1.76.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.76.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - 2x - 2 \)
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The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 4\cdot 23 + 8\cdot 23^{2} + 22\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 6\cdot 23 + 5\cdot 23^{2} + 7\cdot 23^{3} + 15\cdot 23^{4} +O(23^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 11\cdot 23 + 9\cdot 23^{2} + 16\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.