Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(180170657792\)\(\medspace = 2^{12} \cdot 353^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.248438446096.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.1412.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.248438446096.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} + 3x^{5} - 19x^{4} + 41x^{3} - 26x^{2} + 15x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{3} + 9x + 92 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a^{2} + 3 a + 61 + \left(74 a^{2} + 4 a + 93\right)\cdot 97 + \left(59 a^{2} + 22 a + 23\right)\cdot 97^{2} + \left(83 a^{2} + 57 a + 92\right)\cdot 97^{3} + \left(a^{2} + 42 a + 54\right)\cdot 97^{4} + \left(72 a^{2} + 87 a + 41\right)\cdot 97^{5} + \left(32 a^{2} + 44 a + 30\right)\cdot 97^{6} + \left(70 a^{2} + 13 a + 25\right)\cdot 97^{7} + \left(46 a + 32\right)\cdot 97^{8} + \left(58 a^{2} + 46 a + 40\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 45 a^{2} + 62 a + 10 + \left(a^{2} + 46 a + 46\right)\cdot 97 + \left(75 a^{2} + a + 18\right)\cdot 97^{2} + \left(66 a^{2} + 55 a + 88\right)\cdot 97^{3} + \left(78 a^{2} + 93 a + 30\right)\cdot 97^{4} + \left(70 a^{2} + 60 a + 34\right)\cdot 97^{5} + \left(72 a^{2} + 49 a + 76\right)\cdot 97^{6} + \left(7 a^{2} + 77 a + 37\right)\cdot 97^{7} + \left(48 a^{2} + 22 a + 25\right)\cdot 97^{8} + \left(90 a^{2} + 26 a + 41\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 6 + 82\cdot 97 + 33\cdot 97^{2} + 64\cdot 97^{3} + 62\cdot 97^{4} + 7\cdot 97^{5} + 13\cdot 97^{6} + 26\cdot 97^{7} + 13\cdot 97^{8} + 50\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 60 a^{2} + 19 a + 94 + \left(46 a^{2} + 49 a + 38\right)\cdot 97 + \left(20 a^{2} + 86 a + 44\right)\cdot 97^{2} + \left(31 a^{2} + 35 a\right)\cdot 97^{3} + \left(39 a^{2} + 9 a + 42\right)\cdot 97^{4} + \left(2 a^{2} + 40 a + 60\right)\cdot 97^{5} + \left(79 a^{2} + 6 a + 94\right)\cdot 97^{6} + \left(62 a^{2} + 31 a + 7\right)\cdot 97^{7} + \left(31 a^{2} + 45 a + 90\right)\cdot 97^{8} + \left(47 a^{2} + 48 a + 5\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 52 a^{2} + 79 a + 46 + \left(96 a^{2} + 89 a + 47\right)\cdot 97 + \left(13 a^{2} + 2 a + 5\right)\cdot 97^{2} + \left(25 a^{2} + 91 a + 61\right)\cdot 97^{3} + \left(33 a^{2} + 93 a + 5\right)\cdot 97^{4} + \left(80 a^{2} + 14 a + 43\right)\cdot 97^{5} + \left(20 a^{2} + 19 a + 36\right)\cdot 97^{6} + \left(a^{2} + 45 a + 26\right)\cdot 97^{7} + \left(42 a^{2} + 18 a + 55\right)\cdot 97^{8} + \left(57 a^{2} + 64 a + 66\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 82 a^{2} + 96 a + 32 + \left(50 a^{2} + 54 a + 64\right)\cdot 97 + \left(62 a^{2} + 7 a + 5\right)\cdot 97^{2} + \left(40 a^{2} + 67 a + 57\right)\cdot 97^{3} + \left(24 a^{2} + 90 a + 49\right)\cdot 97^{4} + \left(14 a^{2} + 41 a + 34\right)\cdot 97^{5} + \left(94 a^{2} + 71 a + 88\right)\cdot 97^{6} + \left(32 a^{2} + 20 a + 22\right)\cdot 97^{7} + \left(23 a^{2} + 33 a + 40\right)\cdot 97^{8} + \left(89 a^{2} + 81 a + 63\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 24 + 43\cdot 97 + 41\cdot 97^{2} + 75\cdot 97^{3} + 96\cdot 97^{4} + 55\cdot 97^{5} + 71\cdot 97^{6} + 39\cdot 97^{7} + 8\cdot 97^{8} + 58\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 47 a^{2} + 32 a + 22 + \left(21 a^{2} + 46 a + 69\right)\cdot 97 + \left(59 a^{2} + 73 a + 20\right)\cdot 97^{2} + \left(43 a^{2} + 81 a + 46\right)\cdot 97^{3} + \left(16 a^{2} + 57 a + 45\right)\cdot 97^{4} + \left(51 a^{2} + 45 a + 13\right)\cdot 97^{5} + \left(88 a^{2} + 2 a + 74\right)\cdot 97^{6} + \left(18 a^{2} + 6 a + 7\right)\cdot 97^{7} + \left(48 a^{2} + 28 a + 26\right)\cdot 97^{8} + \left(45 a^{2} + 24 a + 62\right)\cdot 97^{9} +O(97^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(4,6)(5,7)$ | $-3$ |
| $9$ | $2$ | $(1,3)(2,8)(4,6)(5,7)$ | $1$ |
| $12$ | $2$ | $(1,2)$ | $3$ |
| $24$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(4,5)$ | $1$ |
| $36$ | $2$ | $(1,2)(4,6)(5,7)$ | $-1$ |
| $16$ | $3$ | $(1,3,8)$ | $0$ |
| $64$ | $3$ | $(1,3,8)(5,6,7)$ | $0$ |
| $12$ | $4$ | $(4,5,6,7)$ | $-3$ |
| $36$ | $4$ | $(1,2,3,8)(4,5,6,7)$ | $1$ |
| $36$ | $4$ | $(1,2,3,8)(4,6)(5,7)$ | $1$ |
| $72$ | $4$ | $(1,4,3,6)(2,5,8,7)$ | $-1$ |
| $72$ | $4$ | $(1,2)(4,5,6,7)$ | $-1$ |
| $144$ | $4$ | $(1,5,2,4)(3,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,8,3)(4,6)(5,7)$ | $0$ |
| $96$ | $6$ | $(1,2)(5,7,6)$ | $0$ |
| $192$ | $6$ | $(1,5,3,6,8,7)(2,4)$ | $0$ |
| $144$ | $8$ | $(1,4,2,5,3,6,8,7)$ | $-1$ |
| $96$ | $12$ | $(1,3,8)(4,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.