Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(115\!\cdots\!288\)\(\medspace = 2^{65} \cdot 3^{22} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.293534171136.7 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | even |
| Determinant: | 1.8.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.293534171136.7 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} - 16x^{5} + 6x^{4} + 96x^{3} + 136x^{2} + 144x + 54 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 5 a^{2} + 18 a + 10 + \left(18 a^{2} + 10 a + 15\right)\cdot 23 + \left(12 a^{2} + 17 a + 2\right)\cdot 23^{2} + \left(20 a^{2} + 15 a + 6\right)\cdot 23^{3} + \left(16 a^{2} + 17 a + 11\right)\cdot 23^{4} + \left(10 a^{2} + 8\right)\cdot 23^{5} + \left(17 a^{2} + 20 a + 1\right)\cdot 23^{6} + \left(22 a^{2} + 11 a + 22\right)\cdot 23^{7} + \left(12 a^{2} + 4 a + 3\right)\cdot 23^{8} + \left(8 a^{2} + 15\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a^{2} + 13 a + 9 + \left(5 a^{2} + 9 a + 17\right)\cdot 23 + \left(14 a^{2} + 8\right)\cdot 23^{2} + \left(19 a^{2} + 5 a + 18\right)\cdot 23^{3} + \left(8 a^{2} + 5 a + 20\right)\cdot 23^{4} + \left(11 a^{2} + 3 a + 20\right)\cdot 23^{5} + \left(8 a^{2} + 19 a + 6\right)\cdot 23^{6} + \left(10 a^{2} + 3 a + 21\right)\cdot 23^{7} + \left(20 a^{2} + 5 a + 10\right)\cdot 23^{8} + \left(21 a^{2} + 10 a + 19\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 11 + \left(18 a^{2} + 10 a + 7\right)\cdot 23 + \left(22 a^{2} + 3 a + 8\right)\cdot 23^{2} + \left(20 a^{2} + 11 a + 14\right)\cdot 23^{3} + \left(2 a^{2} + 15\right)\cdot 23^{4} + \left(12 a^{2} + 11 a + 2\right)\cdot 23^{5} + \left(a^{2} + 17 a + 3\right)\cdot 23^{6} + \left(19 a^{2} + 5 a + 17\right)\cdot 23^{7} + \left(4 a^{2} + 7 a\right)\cdot 23^{8} + \left(16 a^{2} + 11 a + 10\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 + 3\cdot 23 + 20\cdot 23^{2} + 17\cdot 23^{3} + 10\cdot 23^{4} + 17\cdot 23^{5} + 19\cdot 23^{6} + 23^{7} + 17\cdot 23^{8} + 11\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a^{2} + a + 15 + \left(2 a^{2} + 10 a + 6\right)\cdot 23 + \left(21 a^{2} + 9 a + 10\right)\cdot 23^{2} + \left(15 a^{2} + 21 a + 13\right)\cdot 23^{3} + \left(5 a^{2} + 5 a + 16\right)\cdot 23^{4} + \left(10 a^{2} + 3 a + 11\right)\cdot 23^{5} + \left(5 a^{2} + 17 a + 10\right)\cdot 23^{6} + \left(20 a^{2} + 21 a + 11\right)\cdot 23^{7} + \left(20 a^{2} + 21 a + 11\right)\cdot 23^{8} + \left(22 a^{2} + 20 a + 5\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a^{2} + 9 a + 10 + \left(14 a^{2} + 3 a + 7\right)\cdot 23 + \left(10 a^{2} + 13 a + 19\right)\cdot 23^{2} + \left(10 a^{2} + 19 a + 13\right)\cdot 23^{3} + \left(8 a^{2} + 11 a + 12\right)\cdot 23^{4} + \left(a^{2} + 16 a + 7\right)\cdot 23^{5} + \left(9 a^{2} + 9 a + 15\right)\cdot 23^{6} + \left(15 a^{2} + 20 a + 12\right)\cdot 23^{7} + \left(4 a^{2} + 18 a + 20\right)\cdot 23^{8} + \left(a^{2} + 14 a + 14\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 18 a^{2} + 15 a + 12 + \left(9 a^{2} + a + 19\right)\cdot 23 + \left(10 a^{2} + 2 a + 14\right)\cdot 23^{2} + \left(4 a^{2} + 19 a + 7\right)\cdot 23^{3} + \left(3 a^{2} + 4 a + 8\right)\cdot 23^{4} + \left(11 a + 17\right)\cdot 23^{5} + \left(4 a^{2} + 8 a + 21\right)\cdot 23^{6} + \left(4 a^{2} + 5 a + 4\right)\cdot 23^{7} + \left(5 a^{2} + 11 a + 1\right)\cdot 23^{8} + \left(21 a^{2} + 11 a + 9\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 + 14\cdot 23 + 7\cdot 23^{2} + 19\cdot 23^{4} + 5\cdot 23^{5} + 13\cdot 23^{6} + 3\cdot 23^{8} + 6\cdot 23^{9} +O(23^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,6)(5,8)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $2$ |
| $12$ | $2$ | $(1,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,5)$ | $-2$ |
| $36$ | $2$ | $(1,3)(2,6)(5,8)$ | $0$ |
| $16$ | $3$ | $(1,4,7)$ | $0$ |
| $64$ | $3$ | $(1,4,7)(5,6,8)$ | $0$ |
| $12$ | $4$ | $(2,5,6,8)$ | $0$ |
| $36$ | $4$ | $(1,3,4,7)(2,5,6,8)$ | $-2$ |
| $36$ | $4$ | $(1,3,4,7)(2,6)(5,8)$ | $0$ |
| $72$ | $4$ | $(1,2,4,6)(3,5,7,8)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,5,6,8)$ | $2$ |
| $144$ | $4$ | $(1,5,3,2)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,7,4)(2,6)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,3)(5,8,6)$ | $0$ |
| $192$ | $6$ | $(1,5,4,6,7,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,2,3,5,4,6,7,8)$ | $0$ |
| $96$ | $12$ | $(1,4,7)(2,5,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.