Basic invariants
| Dimension: | $7$ |
| Group: | $C_2^3:\GL(3,2)$ |
| Conductor: | \(132705746944\)\(\medspace = 2^{16} \cdot 1423^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.8.132705746944.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2^3:\GL(3,2)$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^3:\GL(3,2)$ |
| Projective stem field: | Galois closure of 8.8.132705746944.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 10x^{6} - 4x^{5} + 27x^{4} + 20x^{3} - 10x^{2} - 8x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a^{2} + 10 a + 11 + \left(23 a^{2} + 10 a + 28\right)\cdot 29 + \left(7 a^{2} + 2 a + 2\right)\cdot 29^{2} + \left(14 a^{2} + 17 a + 16\right)\cdot 29^{3} + \left(22 a^{2} + 8 a + 17\right)\cdot 29^{4} + \left(20 a^{2} + 8 a + 14\right)\cdot 29^{5} + \left(6 a^{2} + 16 a + 7\right)\cdot 29^{6} + \left(20 a^{2} + 15 a + 14\right)\cdot 29^{7} + \left(11 a + 26\right)\cdot 29^{8} + \left(27 a^{2} + 9 a + 7\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 a^{2} + 10 a + 17 + \left(4 a^{2} + 8 a + 2\right)\cdot 29 + \left(9 a^{2} + 14 a + 24\right)\cdot 29^{2} + \left(20 a + 16\right)\cdot 29^{3} + \left(9 a^{2} + 13 a + 28\right)\cdot 29^{4} + \left(26 a^{2} + 13 a + 21\right)\cdot 29^{5} + \left(19 a^{2} + 20 a + 5\right)\cdot 29^{6} + \left(28 a^{2} + 7 a + 6\right)\cdot 29^{7} + \left(22 a^{2} + 27 a + 27\right)\cdot 29^{8} + \left(2 a^{2} + 18 a + 23\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 + 8\cdot 29 + 26\cdot 29^{2} + 19\cdot 29^{3} + 29^{4} + 6\cdot 29^{6} + 14\cdot 29^{7} + 24\cdot 29^{8} + 11\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 + 6\cdot 29 + 7\cdot 29^{2} + 19\cdot 29^{3} + 8\cdot 29^{4} + 28\cdot 29^{5} + 7\cdot 29^{6} + 27\cdot 29^{7} + 5\cdot 29^{8} + 27\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 14 a^{2} + 20 a + 22 + \left(6 a^{2} + 27 a + 16\right)\cdot 29 + \left(3 a^{2} + 13 a\right)\cdot 29^{2} + \left(a^{2} + 23 a + 1\right)\cdot 29^{3} + \left(18 a^{2} + 28 a + 4\right)\cdot 29^{4} + \left(2 a^{2} + a + 7\right)\cdot 29^{5} + \left(12 a^{2} + a + 3\right)\cdot 29^{6} + \left(19 a + 9\right)\cdot 29^{7} + \left(24 a^{2} + 14 a + 25\right)\cdot 29^{8} + \left(16 a^{2} + 11 a + 27\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 3 a^{2} + 22 a + 17 + \left(16 a^{2} + a\right)\cdot 29 + \left(27 a^{2} + 15 a + 4\right)\cdot 29^{2} + \left(4 a^{2} + a + 6\right)\cdot 29^{3} + \left(17 a^{2} + 13 a + 22\right)\cdot 29^{4} + \left(22 a^{2} + 25 a + 4\right)\cdot 29^{5} + \left(10 a^{2} + 7 a + 11\right)\cdot 29^{6} + \left(18 a + 28\right)\cdot 29^{7} + \left(8 a^{2} + 24 a + 3\right)\cdot 29^{8} + \left(13 a^{2} + 13 a + 23\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 16 a^{2} + 9 a + 17 + \left(10 a + 16\right)\cdot 29 + \left(12 a^{2} + 12 a + 8\right)\cdot 29^{2} + \left(14 a^{2} + 20 a + 16\right)\cdot 29^{3} + \left(26 a^{2} + 6 a + 3\right)\cdot 29^{4} + \left(10 a^{2} + 7 a + 11\right)\cdot 29^{5} + \left(2 a^{2} + 21 a + 11\right)\cdot 29^{6} + \left(9 a^{2} + 5 a + 28\right)\cdot 29^{7} + \left(5 a^{2} + 19 a + 22\right)\cdot 29^{8} + \left(28 a^{2} + 28\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 a^{2} + 16 a + \left(6 a^{2} + 28 a + 7\right)\cdot 29 + \left(27 a^{2} + 28 a + 13\right)\cdot 29^{2} + \left(22 a^{2} + 3 a + 20\right)\cdot 29^{3} + \left(22 a^{2} + 16 a\right)\cdot 29^{4} + \left(3 a^{2} + a + 28\right)\cdot 29^{5} + \left(6 a^{2} + 20 a + 4\right)\cdot 29^{6} + \left(28 a^{2} + 20 a + 17\right)\cdot 29^{7} + \left(25 a^{2} + 18 a + 8\right)\cdot 29^{8} + \left(27 a^{2} + 3 a + 23\right)\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $7$ |
| $7$ | $2$ | $(1,5)(2,6)(3,4)(7,8)$ | $-1$ |
| $42$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $-1$ |
| $42$ | $2$ | $(3,5)(6,7)$ | $3$ |
| $224$ | $3$ | $(2,6,7)(3,5,4)$ | $1$ |
| $84$ | $4$ | $(1,2,5,6)(3,8,4,7)$ | $-1$ |
| $168$ | $4$ | $(1,6,8,5)(2,3,4,7)$ | $-1$ |
| $168$ | $4$ | $(1,4)(3,6,5,7)$ | $1$ |
| $224$ | $6$ | $(1,8)(2,5,6,4,7,3)$ | $-1$ |
| $192$ | $7$ | $(1,6,3,7,2,8,5)$ | $0$ |
| $192$ | $7$ | $(1,7,5,3,8,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.