Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(135\)\(\medspace = 3^{3} \cdot 5 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.1.135.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.15.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.135.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} + 3x - 1 \)
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The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 17\cdot 47 + 27\cdot 47^{2} + 42\cdot 47^{3} + 3\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 41 + 38\cdot 47 + 43\cdot 47^{2} + 21\cdot 47^{3} + 15\cdot 47^{4} +O(47^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 42 + 37\cdot 47 + 22\cdot 47^{2} + 29\cdot 47^{3} + 27\cdot 47^{4} +O(47^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.