Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(146\!\cdots\!552\)\(\medspace = 2^{59} \cdot 3^{26} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.764411904.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.764411904.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 2x^{6} + 4x^{5} - 7x^{4} - 4x^{3} + 14x^{2} - 8x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 8 a + 8 + \left(27 a + 6\right)\cdot 31 + \left(10 a + 8\right)\cdot 31^{2} + \left(20 a + 5\right)\cdot 31^{3} + \left(30 a + 3\right)\cdot 31^{4} + \left(3 a + 25\right)\cdot 31^{5} + \left(13 a + 20\right)\cdot 31^{6} + \left(13 a + 29\right)\cdot 31^{7} + \left(10 a + 16\right)\cdot 31^{8} + \left(6 a + 25\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 27 + 27\cdot 31 + 11\cdot 31^{2} + 4\cdot 31^{3} + 13\cdot 31^{4} + 30\cdot 31^{5} + 24\cdot 31^{6} + 14\cdot 31^{7} + 13\cdot 31^{8} + 24\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 4 + \left(12 a + 3\right)\cdot 31 + 9 a\cdot 31^{2} + \left(8 a + 14\right)\cdot 31^{3} + \left(25 a + 13\right)\cdot 31^{4} + \left(3 a + 2\right)\cdot 31^{5} + 21 a\cdot 31^{6} + \left(15 a + 18\right)\cdot 31^{7} + \left(27 a + 21\right)\cdot 31^{8} + \left(13 a + 24\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 13\cdot 31 + 2\cdot 31^{2} + 22\cdot 31^{3} + 16\cdot 31^{4} + 17\cdot 31^{5} + 18\cdot 31^{6} + 24\cdot 31^{7} + 20\cdot 31^{8} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 a + 23 + \left(4 a + 28\right)\cdot 31 + \left(24 a + 8\right)\cdot 31^{2} + \left(21 a + 1\right)\cdot 31^{3} + 30\cdot 31^{4} + \left(19 a + 11\right)\cdot 31^{5} + \left(18 a + 26\right)\cdot 31^{6} + \left(8 a + 29\right)\cdot 31^{7} + \left(8 a + 9\right)\cdot 31^{8} + \left(26 a + 22\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 23 a + 24 + \left(3 a + 21\right)\cdot 31 + \left(20 a + 2\right)\cdot 31^{2} + \left(10 a + 4\right)\cdot 31^{3} + 13\cdot 31^{4} + \left(27 a + 2\right)\cdot 31^{5} + \left(17 a + 12\right)\cdot 31^{6} + \left(17 a + 12\right)\cdot 31^{7} + \left(20 a + 24\right)\cdot 31^{8} + \left(24 a + 27\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( a + 21 + \left(26 a + 8\right)\cdot 31 + \left(6 a + 21\right)\cdot 31^{2} + \left(9 a + 20\right)\cdot 31^{3} + \left(30 a + 9\right)\cdot 31^{4} + \left(11 a + 18\right)\cdot 31^{5} + \left(12 a + 13\right)\cdot 31^{6} + \left(22 a + 28\right)\cdot 31^{7} + \left(22 a + 17\right)\cdot 31^{8} + \left(4 a + 4\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 17 a + 1 + \left(18 a + 14\right)\cdot 31 + \left(21 a + 6\right)\cdot 31^{2} + \left(22 a + 21\right)\cdot 31^{3} + \left(5 a + 24\right)\cdot 31^{4} + \left(27 a + 15\right)\cdot 31^{5} + \left(9 a + 7\right)\cdot 31^{6} + \left(15 a + 28\right)\cdot 31^{7} + \left(3 a + 29\right)\cdot 31^{8} + \left(17 a + 24\right)\cdot 31^{9} +O(31^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(3,7)(5,8)$ | $-6$ |
| $9$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $2$ |
| $12$ | $2$ | $(1,2)$ | $0$ |
| $24$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $36$ | $2$ | $(1,2)(3,5)$ | $-2$ |
| $36$ | $2$ | $(1,2)(3,7)(5,8)$ | $0$ |
| $16$ | $3$ | $(1,4,6)$ | $0$ |
| $64$ | $3$ | $(1,4,6)(5,7,8)$ | $0$ |
| $12$ | $4$ | $(3,5,7,8)$ | $0$ |
| $36$ | $4$ | $(1,2,4,6)(3,5,7,8)$ | $-2$ |
| $36$ | $4$ | $(1,2,4,6)(3,7)(5,8)$ | $0$ |
| $72$ | $4$ | $(1,3,4,7)(2,5,6,8)$ | $0$ |
| $72$ | $4$ | $(1,2)(3,5,7,8)$ | $2$ |
| $144$ | $4$ | $(1,5,2,3)(4,7)(6,8)$ | $0$ |
| $48$ | $6$ | $(1,6,4)(3,7)(5,8)$ | $0$ |
| $96$ | $6$ | $(1,2)(5,8,7)$ | $0$ |
| $192$ | $6$ | $(1,5,4,7,6,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,3,2,5,4,7,6,8)$ | $0$ |
| $96$ | $12$ | $(1,4,6)(3,5,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.