Basic invariants
| Dimension: | $6$ |
| Group: | $V_4^2:(S_3\times C_2)$ |
| Conductor: | \(4552163129\)\(\medspace = 41^{3} \cdot 257^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.186638688289.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $V_4^2:(S_3\times C_2)$ |
| Parity: | even |
| Determinant: | 1.41.2t1.a.a |
| Projective image: | $C_2^3:S_4$ |
| Projective stem field: | Galois closure of 8.4.186638688289.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 8x^{6} + 6x^{5} + 17x^{4} - 13x^{3} + x^{2} + 17x - 10 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{3} + 2x + 18 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a^{2} + 3 a + 8 + \left(9 a^{2} + 1\right)\cdot 23 + \left(11 a^{2} + 7 a + 13\right)\cdot 23^{2} + \left(13 a^{2} + 21 a + 13\right)\cdot 23^{3} + \left(15 a^{2} + 4 a + 2\right)\cdot 23^{4} + \left(a^{2} + 14 a + 13\right)\cdot 23^{5} + \left(6 a^{2} + 20 a + 1\right)\cdot 23^{6} + \left(12 a^{2} + 11 a + 13\right)\cdot 23^{7} + \left(4 a^{2} + 4 a + 12\right)\cdot 23^{8} + \left(7 a^{2} + 22\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a^{2} + 6 a + 5 + \left(5 a^{2} + 4 a + 3\right)\cdot 23 + \left(3 a^{2} + 7 a + 2\right)\cdot 23^{2} + \left(15 a + 11\right)\cdot 23^{3} + \left(21 a^{2} + a + 17\right)\cdot 23^{4} + \left(21 a^{2} + 11 a + 1\right)\cdot 23^{5} + \left(2 a^{2} + 10 a + 5\right)\cdot 23^{6} + \left(9 a^{2} + 17 a + 1\right)\cdot 23^{7} + \left(11 a^{2} + 6 a + 14\right)\cdot 23^{8} + \left(20 a^{2} + 9 a + 9\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 8\cdot 23 + 17\cdot 23^{2} + 5\cdot 23^{3} + 22\cdot 23^{4} + 15\cdot 23^{5} + 22\cdot 23^{7} + 15\cdot 23^{8} + 8\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 21 a + 20 + \left(17 a^{2} + 4 a + 6\right)\cdot 23 + \left(6 a^{2} + 17 a + 20\right)\cdot 23^{2} + \left(11 a^{2} + 6 a + 16\right)\cdot 23^{3} + \left(15 a^{2} + 11\right)\cdot 23^{4} + \left(2 a^{2} + 17 a + 21\right)\cdot 23^{5} + \left(2 a^{2} + 11 a + 11\right)\cdot 23^{6} + \left(3 a^{2} + 13 a + 17\right)\cdot 23^{7} + \left(21 a^{2} + 16 a + 6\right)\cdot 23^{8} + \left(21 a^{2} + 4 a + 8\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 12 a^{2} + 14 a + 19 + \left(7 a^{2} + 18 a + 5\right)\cdot 23 + \left(8 a^{2} + 8 a + 1\right)\cdot 23^{2} + \left(9 a^{2} + 9 a + 8\right)\cdot 23^{3} + \left(9 a^{2} + 16 a + 17\right)\cdot 23^{4} + \left(22 a^{2} + 20 a + 17\right)\cdot 23^{5} + \left(13 a^{2} + 14 a + 19\right)\cdot 23^{6} + \left(a^{2} + 16 a + 21\right)\cdot 23^{7} + \left(7 a^{2} + 11 a + 15\right)\cdot 23^{8} + \left(18 a^{2} + 13 a + 6\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a^{2} + 20 a + 9 + \left(11 a^{2} + 4 a + 22\right)\cdot 23 + \left(12 a^{2} + 16 a + 4\right)\cdot 23^{2} + \left(17 a^{2} + 22 a + 2\right)\cdot 23^{3} + \left(8 a^{2} + 4 a + 18\right)\cdot 23^{4} + \left(7 a^{2} + 21 a + 4\right)\cdot 23^{5} + \left(6 a + 17\right)\cdot 23^{6} + \left(3 a^{2} + 12 a + 9\right)\cdot 23^{7} + \left(9 a^{2} + 12 a + 21\right)\cdot 23^{8} + \left(19 a^{2} + 12 a + 4\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 + 6\cdot 23 + 23^{2} + 2\cdot 23^{3} + 13\cdot 23^{4} + 12\cdot 23^{5} + 14\cdot 23^{6} + 16\cdot 23^{7} + 5\cdot 23^{8} + 15\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 2 a^{2} + 5 a + 7 + \left(17 a^{2} + 13 a + 14\right)\cdot 23 + \left(3 a^{2} + 12 a + 8\right)\cdot 23^{2} + \left(17 a^{2} + 16 a + 9\right)\cdot 23^{3} + \left(21 a^{2} + 17 a + 12\right)\cdot 23^{4} + \left(12 a^{2} + 7 a + 4\right)\cdot 23^{5} + \left(20 a^{2} + 4 a + 21\right)\cdot 23^{6} + \left(16 a^{2} + 20 a + 12\right)\cdot 23^{7} + \left(15 a^{2} + 16 a + 22\right)\cdot 23^{8} + \left(4 a^{2} + 5 a + 15\right)\cdot 23^{9} +O(23^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $6$ |
| $3$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $0$ |
| $6$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $-2$ |
| $6$ | $2$ | $(1,5)(2,7)$ | $2$ |
| $12$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $0$ |
| $12$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $32$ | $3$ | $(2,5,7)(3,4,6)$ | $0$ |
| $12$ | $4$ | $(1,8,5,3)(2,4,7,6)$ | $0$ |
| $12$ | $4$ | $(1,3,5,8)(2,4,7,6)$ | $0$ |
| $12$ | $4$ | $(1,5,2,7)(3,6,4,8)$ | $-2$ |
| $24$ | $4$ | $(1,3,7,8)(2,4,5,6)$ | $0$ |
| $24$ | $4$ | $(3,6,4,8)(5,7)$ | $0$ |
| $32$ | $6$ | $(1,8)(2,4,5,6,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.