Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(439\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | 3.1.439.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Determinant: | 1.439.2t1.a.a |
| Projective image: | $S_3$ |
| Projective stem field: | 3.1.439.1 |
Defining polynomial
| $f(x)$ | $=$ | \(x^{3} - x^{2} - 2 x + 5\)  . |
The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ | \( 3\cdot 5 +O(5^{5})\)
|
| $r_{ 2 }$ | $=$ | \( 2 + 4\cdot 5 + 4\cdot 5^{3} + 4\cdot 5^{4} +O(5^{5})\)
|
| $r_{ 3 }$ | $=$ | \( 4 + 2\cdot 5 + 3\cdot 5^{2} +O(5^{5})\)
|
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.