Basic invariants
| Dimension: | $4$ |
| Group: | $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Conductor: | \(4922883\)\(\medspace = 3^{3} \cdot 7^{2} \cdot 61^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.6512974209.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $(((C_4 \times C_2): C_2):C_2):C_2$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $C_2^2\wr C_2$ |
| Projective stem field: | Galois closure of 8.0.132917841.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - 9x^{6} + 8x^{5} + 25x^{4} - 24x^{3} - 18x^{2} + 30x - 3 \)
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The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 a + 19 + 7\cdot 31 + \left(13 a + 11\right)\cdot 31^{2} + \left(10 a + 3\right)\cdot 31^{3} + \left(21 a + 28\right)\cdot 31^{4} + \left(26 a + 6\right)\cdot 31^{5} + \left(19 a + 2\right)\cdot 31^{6} + \left(14 a + 19\right)\cdot 31^{7} + \left(30 a + 13\right)\cdot 31^{8} + \left(25 a + 13\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 23 a + 1 + \left(24 a + 14\right)\cdot 31 + \left(10 a + 20\right)\cdot 31^{2} + \left(12 a + 25\right)\cdot 31^{3} + \left(20 a + 11\right)\cdot 31^{4} + \left(30 a + 10\right)\cdot 31^{5} + \left(2 a + 30\right)\cdot 31^{6} + 29 a\cdot 31^{7} + \left(7 a + 19\right)\cdot 31^{8} + \left(18 a + 1\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a + 6 + \left(17 a + 7\right)\cdot 31 + \left(6 a + 20\right)\cdot 31^{2} + \left(a + 21\right)\cdot 31^{3} + \left(8 a + 9\right)\cdot 31^{4} + \left(27 a + 27\right)\cdot 31^{5} + \left(22 a + 2\right)\cdot 31^{6} + \left(13 a + 12\right)\cdot 31^{7} + \left(16 a + 9\right)\cdot 31^{8} + \left(17 a + 12\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 16 + \left(6 a + 9\right)\cdot 31 + \left(20 a + 17\right)\cdot 31^{2} + \left(18 a + 8\right)\cdot 31^{3} + \left(10 a + 9\right)\cdot 31^{4} + 20\cdot 31^{5} + \left(28 a + 5\right)\cdot 31^{6} + \left(a + 25\right)\cdot 31^{7} + \left(23 a + 5\right)\cdot 31^{8} + \left(12 a + 30\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a + 5 + \left(13 a + 27\right)\cdot 31 + \left(24 a + 15\right)\cdot 31^{2} + \left(29 a + 17\right)\cdot 31^{3} + \left(22 a + 24\right)\cdot 31^{4} + \left(3 a + 11\right)\cdot 31^{5} + \left(8 a + 21\right)\cdot 31^{6} + \left(17 a + 16\right)\cdot 31^{7} + \left(14 a + 28\right)\cdot 31^{8} + \left(13 a + 30\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a + 19 + \left(15 a + 22\right)\cdot 31 + \left(12 a + 11\right)\cdot 31^{2} + \left(22 a + 17\right)\cdot 31^{3} + \left(8 a + 23\right)\cdot 31^{4} + 23 a\cdot 31^{5} + \left(24 a + 10\right)\cdot 31^{6} + \left(12 a + 10\right)\cdot 31^{7} + \left(26 a + 20\right)\cdot 31^{8} + \left(25 a + 18\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 23 a + 4 + \left(15 a + 14\right)\cdot 31 + \left(18 a + 21\right)\cdot 31^{2} + \left(8 a + 18\right)\cdot 31^{3} + \left(22 a + 18\right)\cdot 31^{4} + \left(7 a + 7\right)\cdot 31^{5} + \left(6 a + 5\right)\cdot 31^{6} + \left(18 a + 11\right)\cdot 31^{7} + \left(4 a + 29\right)\cdot 31^{8} + \left(5 a + 12\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 13 a + 24 + \left(30 a + 21\right)\cdot 31 + \left(17 a + 5\right)\cdot 31^{2} + \left(20 a + 11\right)\cdot 31^{3} + \left(9 a + 29\right)\cdot 31^{4} + \left(4 a + 7\right)\cdot 31^{5} + \left(11 a + 15\right)\cdot 31^{6} + \left(16 a + 28\right)\cdot 31^{7} + 28\cdot 31^{8} + \left(5 a + 3\right)\cdot 31^{9} +O(31^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $-4$ |
| $2$ | $2$ | $(2,8)(3,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,6)(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,4)(3,6)$ | $0$ |
| $4$ | $2$ | $(2,3)(6,8)$ | $2$ |
| $4$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $4$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $0$ |
| $4$ | $4$ | $(1,6,4,3)(2,7,8,5)$ | $0$ |
| $4$ | $4$ | $(1,5,4,7)(2,3,8,6)$ | $0$ |
| $4$ | $4$ | $(1,6,4,3)(2,5,8,7)$ | $0$ |
| $8$ | $4$ | $(1,4)(2,3,8,6)$ | $0$ |
| $8$ | $4$ | $(1,6,5,2)(3,7,8,4)$ | $0$ |
| $8$ | $4$ | $(1,6,5,8)(2,4,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.