Basic invariants
| Dimension: | $4$ |
| Group: | $C_2^3:S_4$ |
| Conductor: | \(7537\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.56806369.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $C_2^3:S_4$ |
| Parity: | even |
| Determinant: | 1.7537.2t1.a.a |
| Projective image: | $C_2^2:S_4$ |
| Projective stem field: | Galois closure of 8.0.3226963558964161.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - x^{7} - x^{6} + x^{5} - x^{4} + x^{3} - x^{2} - x + 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 12 a^{2} + 11 + \left(14 a^{2} + 4 a + 4\right)\cdot 23 + \left(19 a^{2} + 11 a + 14\right)\cdot 23^{2} + \left(7 a^{2} + 4 a + 18\right)\cdot 23^{3} + \left(9 a^{2} + 17 a + 5\right)\cdot 23^{4} + \left(13 a^{2} + 17 a + 19\right)\cdot 23^{5} + \left(20 a^{2} + 9 a\right)\cdot 23^{6} + \left(19 a^{2} + 20 a + 13\right)\cdot 23^{7} + \left(18 a^{2} + 11 a\right)\cdot 23^{8} + \left(15 a^{2} + 13 a + 18\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 + 23 + 3\cdot 23^{3} + 15\cdot 23^{4} + 10\cdot 23^{5} + 6\cdot 23^{6} + 8\cdot 23^{7} + 22\cdot 23^{8} + 2\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a^{2} + 15 a + 8 + \left(2 a^{2} + 2 a + 10\right)\cdot 23 + \left(14 a^{2} + 21 a + 5\right)\cdot 23^{2} + \left(10 a^{2} + 13 a + 8\right)\cdot 23^{3} + \left(14 a + 18\right)\cdot 23^{4} + \left(22 a^{2} + 16 a + 4\right)\cdot 23^{5} + \left(a^{2} + 4 a + 21\right)\cdot 23^{6} + \left(12 a^{2} + 7 a + 4\right)\cdot 23^{7} + \left(21 a^{2} + 9 a + 5\right)\cdot 23^{8} + \left(6 a^{2} + 8 a + 14\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a^{2} + 19 a + 17 + \left(22 a^{2} + 5 a + 14\right)\cdot 23 + \left(16 a^{2} + 15 a + 10\right)\cdot 23^{2} + \left(15 a^{2} + 17 a + 21\right)\cdot 23^{3} + \left(8 a^{2} + 12 a + 4\right)\cdot 23^{4} + \left(20 a^{2} + 9 a + 13\right)\cdot 23^{5} + \left(22 a^{2} + 12 a + 11\right)\cdot 23^{6} + \left(5 a^{2} + a + 17\right)\cdot 23^{7} + \left(11 a^{2} + 4 a + 5\right)\cdot 23^{8} + \left(10 a^{2} + 21 a + 3\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a^{2} + 16 a + 10 + \left(15 a^{2} + 22 a + 19\right)\cdot 23 + \left(6 a^{2} + 3 a + 10\right)\cdot 23^{2} + \left(14 a^{2} + 4 a + 5\right)\cdot 23^{3} + \left(10 a^{2} + 13 a + 1\right)\cdot 23^{4} + \left(8 a^{2} + 19 a + 2\right)\cdot 23^{5} + \left(18 a^{2} + 4 a + 20\right)\cdot 23^{6} + \left(9 a^{2} + 14 a + 1\right)\cdot 23^{7} + \left(4 a^{2} + 6 a + 13\right)\cdot 23^{8} + \left(9 a^{2} + 19 a + 9\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a^{2} + 4 a + 3 + \left(9 a^{2} + 13 a + 5\right)\cdot 23 + \left(9 a^{2} + 19 a + 8\right)\cdot 23^{2} + \left(22 a^{2} + 7\right)\cdot 23^{3} + \left(4 a^{2} + 16 a + 15\right)\cdot 23^{4} + \left(12 a^{2} + 18 a + 17\right)\cdot 23^{5} + \left(2 a^{2} + 22\right)\cdot 23^{6} + \left(20 a^{2} + a + 20\right)\cdot 23^{7} + \left(15 a^{2} + 7 a + 11\right)\cdot 23^{8} + \left(19 a^{2} + 11 a + 15\right)\cdot 23^{9} +O(23^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 7 + 22\cdot 23 + 6\cdot 23^{2} + 13\cdot 23^{3} + 20\cdot 23^{4} + 12\cdot 23^{5} + 17\cdot 23^{6} + 19\cdot 23^{7} + 6\cdot 23^{8} + 14\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 9 a^{2} + 15 a + 4 + \left(5 a^{2} + 20 a + 14\right)\cdot 23 + \left(2 a^{2} + 20 a + 12\right)\cdot 23^{2} + \left(21 a^{2} + 4 a + 14\right)\cdot 23^{3} + \left(11 a^{2} + 18 a + 10\right)\cdot 23^{4} + \left(15 a^{2} + 9 a + 11\right)\cdot 23^{5} + \left(2 a^{2} + 13 a + 14\right)\cdot 23^{6} + \left(a^{2} + a + 5\right)\cdot 23^{7} + \left(20 a^{2} + 7 a + 3\right)\cdot 23^{8} + \left(6 a^{2} + 18 a + 14\right)\cdot 23^{9} +O(23^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $6$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $6$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $6$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $12$ | $2$ | $(1,3)(6,8)$ | $2$ |
| $12$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $-2$ |
| $32$ | $3$ | $(2,3,4)(5,7,6)$ | $1$ |
| $12$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $24$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $0$ |
| $24$ | $4$ | $(1,6,5,7)(2,8,3,4)$ | $0$ |
| $24$ | $4$ | $(1,3,8,6)(2,7)$ | $0$ |
| $32$ | $6$ | $(1,4,2,8,5,7)(3,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.