Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(524\)\(\medspace = 2^{2} \cdot 131 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 3.1.524.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.131.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | Galois closure of 3.1.524.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{3} - x^{2} + 3x - 5 \)
|
The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2\cdot 5 + 5^{2} + 2\cdot 5^{3} +O(5^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 4\cdot 5 + 5^{2} + 5^{3} +O(5^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 + 3\cdot 5 + 5^{2} + 5^{3} + 4\cdot 5^{4} +O(5^{5})\)
|
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.