Basic invariants
| Dimension: | $2$ |
| Group: | $D_{6}$ |
| Conductor: | \(1176\)\(\medspace = 2^{3} \cdot 3 \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 6.0.9680832.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{6}$ |
| Parity: | odd |
| Projective image: | $S_3$ |
| Projective field: | Galois closure of 3.1.1176.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{2} + 21x + 5 \)
![Copy content]()
![Toggle raw display]()
Roots:
| $r_{ 1 }$ | $=$ |
\( 17 a + 1 + \left(a + 11\right)\cdot 23 + \left(4 a + 10\right)\cdot 23^{2} + \left(11 a + 16\right)\cdot 23^{3} + \left(12 a + 21\right)\cdot 23^{4} + \left(7 a + 22\right)\cdot 23^{5} + \left(5 a + 9\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 15\cdot 23 + 7\cdot 23^{2} + 15\cdot 23^{3} + 5\cdot 23^{4} + 5\cdot 23^{5} + 17\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 + 7\cdot 23 + 15\cdot 23^{2} + 7\cdot 23^{3} + 17\cdot 23^{4} + 17\cdot 23^{5} + 5\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a + 12 + \left(21 a + 20\right)\cdot 23 + \left(18 a + 16\right)\cdot 23^{2} + \left(11 a + 11\right)\cdot 23^{3} + \left(10 a + 12\right)\cdot 23^{4} + \left(15 a + 2\right)\cdot 23^{5} + \left(17 a + 13\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a + \left(21 a + 12\right)\cdot 23 + \left(18 a + 12\right)\cdot 23^{2} + \left(11 a + 6\right)\cdot 23^{3} + \left(10 a + 1\right)\cdot 23^{4} + 15 a\cdot 23^{5} + \left(17 a + 13\right)\cdot 23^{6} +O(23^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + 12 + \left(a + 2\right)\cdot 23 + \left(4 a + 6\right)\cdot 23^{2} + \left(11 a + 11\right)\cdot 23^{3} + \left(12 a + 10\right)\cdot 23^{4} + \left(7 a + 20\right)\cdot 23^{5} + \left(5 a + 9\right)\cdot 23^{6} +O(23^{7})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,4,2)(3,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,6,2,5,4,3)$ | $1$ |