Basic invariants
| Dimension: | $2$ |
| Group: | $S_3$ |
| Conductor: | \(23\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 3.1.23.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_3$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.23.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 + 39\cdot 59 + 39\cdot 59^{2} + 36\cdot 59^{3} + 59^{4} +O(59^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 9 + 59 + 51\cdot 59^{2} + 39\cdot 59^{3} + 51\cdot 59^{4} +O(59^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 44 + 18\cdot 59 + 27\cdot 59^{2} + 41\cdot 59^{3} + 5\cdot 59^{4} +O(59^{5})\)
| |
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $3$ | $(1,2,3)$ | $-1$ |