Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(463\!\cdots\!152\)\(\medspace = 2^{67} \cdot 3^{22} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.587068342272.1 |
| 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 8.4.587068342272.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} - 8x^{5} + 36x^{4} + 48x^{3} + 16x^{2} - 6 \)
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The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 14\cdot 53 + 28\cdot 53^{2} + 43\cdot 53^{3} + 8\cdot 53^{4} + 23\cdot 53^{5} + 41\cdot 53^{6} + 20\cdot 53^{7} + 29\cdot 53^{8} + 38\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 a + 32 + \left(41 a + 7\right)\cdot 53 + 48 a\cdot 53^{2} + \left(48 a + 12\right)\cdot 53^{3} + \left(3 a + 11\right)\cdot 53^{4} + \left(50 a + 49\right)\cdot 53^{5} + \left(24 a + 10\right)\cdot 53^{6} + \left(24 a + 20\right)\cdot 53^{7} + \left(5 a + 7\right)\cdot 53^{8} + \left(25 a + 28\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 a + 28 + \left(48 a + 50\right)\cdot 53 + \left(21 a + 40\right)\cdot 53^{2} + \left(47 a + 25\right)\cdot 53^{3} + \left(14 a + 15\right)\cdot 53^{4} + \left(30 a + 8\right)\cdot 53^{5} + \left(32 a + 41\right)\cdot 53^{6} + \left(38 a + 7\right)\cdot 53^{7} + \left(52 a + 19\right)\cdot 53^{8} + \left(33 a + 26\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 48 + 37\cdot 53 + 29\cdot 53^{2} + 50\cdot 53^{3} + 53^{4} + 46\cdot 53^{6} + 24\cdot 53^{7} + 11\cdot 53^{8} + 22\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 32 a + 10 + \left(11 a + 46\right)\cdot 53 + \left(4 a + 47\right)\cdot 53^{2} + \left(4 a + 52\right)\cdot 53^{3} + \left(49 a + 30\right)\cdot 53^{4} + \left(2 a + 33\right)\cdot 53^{5} + \left(28 a + 7\right)\cdot 53^{6} + \left(28 a + 40\right)\cdot 53^{7} + \left(47 a + 4\right)\cdot 53^{8} + \left(27 a + 17\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 48 a + 38 + \left(38 a + 49\right)\cdot 53 + \left(16 a + 4\right)\cdot 53^{2} + \left(17 a + 23\right)\cdot 53^{3} + \left(23 a + 46\right)\cdot 53^{4} + \left(26 a + 29\right)\cdot 53^{5} + \left(9 a + 46\right)\cdot 53^{7} + \left(36 a + 38\right)\cdot 53^{8} + \left(8 a + 38\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 28 a + 22 + \left(4 a + 7\right)\cdot 53 + \left(31 a + 27\right)\cdot 53^{2} + \left(5 a + 34\right)\cdot 53^{3} + \left(38 a + 27\right)\cdot 53^{4} + \left(22 a + 8\right)\cdot 53^{5} + \left(20 a + 35\right)\cdot 53^{6} + \left(14 a + 23\right)\cdot 53^{7} + 32\cdot 53^{8} + \left(19 a + 3\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 5 a + 18 + \left(14 a + 51\right)\cdot 53 + \left(36 a + 32\right)\cdot 53^{2} + \left(35 a + 22\right)\cdot 53^{3} + \left(29 a + 16\right)\cdot 53^{4} + \left(26 a + 6\right)\cdot 53^{5} + \left(52 a + 29\right)\cdot 53^{6} + \left(43 a + 28\right)\cdot 53^{7} + \left(16 a + 15\right)\cdot 53^{8} + \left(44 a + 37\right)\cdot 53^{9} +O(53^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(3,7)(6,8)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,5)(3,7)(6,8)$ | $2$ |
| $12$ | $2$ | $(1,2)$ | $0$ |
| $24$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $0$ |
| $36$ | $2$ | $(1,2)(3,8)$ | $-2$ |
| $36$ | $2$ | $(1,2)(3,7)(6,8)$ | $0$ |
| $16$ | $3$ | $(2,5,4)$ | $0$ |
| $64$ | $3$ | $(2,5,4)(6,8,7)$ | $0$ |
| $12$ | $4$ | $(3,6,7,8)$ | $0$ |
| $36$ | $4$ | $(1,2,4,5)(3,6,7,8)$ | $-2$ |
| $36$ | $4$ | $(1,2,4,5)(3,7)(6,8)$ | $0$ |
| $72$ | $4$ | $(1,3,4,7)(2,6,5,8)$ | $0$ |
| $72$ | $4$ | $(1,2)(3,6,7,8)$ | $2$ |
| $144$ | $4$ | $(1,3,2,8)(4,6)(5,7)$ | $0$ |
| $48$ | $6$ | $(2,4,5)(3,7)(6,8)$ | $0$ |
| $96$ | $6$ | $(2,5,4)(3,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,8,5,7,4,6)$ | $0$ |
| $144$ | $8$ | $(1,3,2,6,4,7,5,8)$ | $0$ |
| $96$ | $12$ | $(2,5,4)(3,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.