Basic invariants
| Dimension: | $6$ |
| Group: | $V_4^2:(S_3\times C_2)$ |
| Conductor: | \(21367509450125\)\(\medspace = 5^{3} \cdot 643^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.4.258405625.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T108 |
| Parity: | even |
| Projective image: | $C_2^3:S_4$ |
| Projective field: | Galois closure of 8.4.258405625.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a^{2} + 12 a + 19 + \left(4 a^{2} + 22 a + 22\right)\cdot 29 + \left(25 a^{2} + 23 a + 8\right)\cdot 29^{2} + \left(9 a^{2} + 14 a + 9\right)\cdot 29^{3} + \left(13 a^{2} + 10 a + 23\right)\cdot 29^{4} + \left(9 a^{2} + 4 a + 9\right)\cdot 29^{5} + \left(18 a^{2} + 25 a + 20\right)\cdot 29^{6} + \left(12 a^{2} + 11\right)\cdot 29^{7} + \left(18 a^{2} + 21 a + 15\right)\cdot 29^{8} + \left(25 a^{2} + 6 a\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 a^{2} + 7 a + 2 + \left(5 a^{2} + 26 a + 5\right)\cdot 29 + \left(6 a^{2} + 13 a + 22\right)\cdot 29^{2} + \left(3 a^{2} + 22 a + 19\right)\cdot 29^{3} + \left(23 a^{2} + 2 a + 26\right)\cdot 29^{4} + \left(25 a^{2} + 14 a + 21\right)\cdot 29^{5} + \left(21 a^{2} + a + 5\right)\cdot 29^{6} + \left(23 a^{2} + 14 a + 7\right)\cdot 29^{7} + \left(10 a^{2} + 20 a + 5\right)\cdot 29^{8} + \left(5 a^{2} + 20 a + 12\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a + 28 + \left(9 a^{2} + 3 a + 13\right)\cdot 29 + \left(20 a^{2} + 19 a + 25\right)\cdot 29^{2} + \left(19 a^{2} + 18 a + 18\right)\cdot 29^{3} + \left(20 a + 24\right)\cdot 29^{4} + \left(27 a^{2} + 14 a + 4\right)\cdot 29^{5} + \left(22 a^{2} + 12 a\right)\cdot 29^{6} + \left(8 a^{2} + 13 a + 22\right)\cdot 29^{7} + \left(26 a^{2} + 26 a + 23\right)\cdot 29^{8} + \left(22 a^{2} + 17 a + 8\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a^{2} + 11 + \left(2 a^{2} + 3 a + 24\right)\cdot 29 + \left(9 a^{2} + 9 a\right)\cdot 29^{2} + \left(22 a^{2} + 15 a + 3\right)\cdot 29^{3} + \left(7 a^{2} + 8 a + 5\right)\cdot 29^{4} + \left(10 a^{2} + 20 a + 21\right)\cdot 29^{5} + \left(8 a^{2} + 9 a + 9\right)\cdot 29^{6} + \left(11 a^{2} + 6 a + 25\right)\cdot 29^{7} + \left(10 a^{2} + 25 a + 21\right)\cdot 29^{8} + \left(5 a^{2} + 11 a + 4\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a^{2} + 2 a + 16 + \left(17 a^{2} + 22 a + 25\right)\cdot 29 + \left(28 a^{2} + 26\right)\cdot 29^{2} + \left(15 a^{2} + 24 a + 13\right)\cdot 29^{3} + \left(20 a^{2} + 28 a + 12\right)\cdot 29^{4} + \left(20 a^{2} + 22 a + 25\right)\cdot 29^{5} + \left(26 a^{2} + 6 a + 14\right)\cdot 29^{6} + \left(8 a^{2} + 9 a + 12\right)\cdot 29^{7} + \left(21 a^{2} + 6 a + 7\right)\cdot 29^{8} + \left(28 a + 8\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 14 + 21\cdot 29^{2} + 10\cdot 29^{3} + 27\cdot 29^{4} + 9\cdot 29^{5} + 11\cdot 29^{6} + 15\cdot 29^{7} + 5\cdot 29^{8} + 23\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 + 29 + 23\cdot 29^{3} + 5\cdot 29^{5} + 5\cdot 29^{6} + 27\cdot 29^{7} + 26\cdot 29^{8} + 26\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 17 a^{2} + 10 a + 1 + \left(18 a^{2} + 9 a + 22\right)\cdot 29 + \left(26 a^{2} + 20 a + 10\right)\cdot 29^{2} + \left(15 a^{2} + 20 a + 17\right)\cdot 29^{3} + \left(21 a^{2} + 15 a + 24\right)\cdot 29^{4} + \left(22 a^{2} + 10 a + 17\right)\cdot 29^{5} + \left(17 a^{2} + 2 a + 19\right)\cdot 29^{6} + \left(21 a^{2} + 14 a + 23\right)\cdot 29^{7} + \left(28 a^{2} + 16 a + 9\right)\cdot 29^{8} + \left(26 a^{2} + a + 2\right)\cdot 29^{9} +O(29^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $6$ |
| $3$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $-2$ |
| $4$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $6$ | $2$ | $(1,2)(6,8)$ | $2$ |
| $6$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ |
| $12$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $0$ |
| $12$ | $2$ | $(5,7)(6,8)$ | $-2$ |
| $32$ | $3$ | $(1,6,2)(3,7,5)$ | $0$ |
| $12$ | $4$ | $(1,7,6,4)(2,3,8,5)$ | $0$ |
| $12$ | $4$ | $(1,8,2,6)(3,5,4,7)$ | $2$ |
| $12$ | $4$ | $(1,7,6,4)(2,5,8,3)$ | $0$ |
| $24$ | $4$ | $(1,5,2,4)(3,8,7,6)$ | $0$ |
| $24$ | $4$ | $(1,8,2,6)(5,7)$ | $0$ |
| $32$ | $6$ | $(1,4)(2,5,6,3,8,7)$ | $0$ |