Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(830\!\cdots\!552\)\(\medspace = 2^{62} \cdot 23^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.25516048384.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | even |
| Determinant: | 1.92.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.25516048384.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 6x^{6} - 20x^{5} + 23x^{4} + 28x^{3} + 8x^{2} - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{2} + 96x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 33 a + 14 + \left(57 a + 25\right)\cdot 97 + \left(5 a + 14\right)\cdot 97^{2} + \left(53 a + 72\right)\cdot 97^{3} + \left(57 a + 5\right)\cdot 97^{4} + \left(60 a + 50\right)\cdot 97^{5} + \left(5 a + 43\right)\cdot 97^{6} + \left(42 a + 41\right)\cdot 97^{7} + \left(55 a + 84\right)\cdot 97^{8} + \left(66 a + 22\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 53 + 7\cdot 97 + 76\cdot 97^{2} + 69\cdot 97^{3} + 48\cdot 97^{4} + 96\cdot 97^{5} + 62\cdot 97^{6} + 46\cdot 97^{7} + 20\cdot 97^{8} + 53\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 86 + 84\cdot 97 + 57\cdot 97^{2} + 54\cdot 97^{3} + 37\cdot 97^{4} + 23\cdot 97^{5} + 72\cdot 97^{6} + 33\cdot 97^{7} + 82\cdot 97^{8} + 60\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 76 a + 24 + \left(29 a + 32\right)\cdot 97 + \left(49 a + 39\right)\cdot 97^{2} + \left(33 a + 83\right)\cdot 97^{3} + 37\cdot 97^{4} + \left(28 a + 27\right)\cdot 97^{5} + \left(75 a + 73\right)\cdot 97^{6} + \left(2 a + 36\right)\cdot 97^{7} + \left(20 a + 93\right)\cdot 97^{8} + \left(58 a + 14\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 a + 47 + \left(39 a + 49\right)\cdot 97 + \left(91 a + 59\right)\cdot 97^{2} + \left(43 a + 22\right)\cdot 97^{3} + \left(39 a + 10\right)\cdot 97^{4} + \left(36 a + 53\right)\cdot 97^{5} + \left(91 a + 85\right)\cdot 97^{6} + \left(54 a + 77\right)\cdot 97^{7} + 41 a\cdot 97^{8} + \left(30 a + 34\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 21 a + 3 + \left(67 a + 83\right)\cdot 97 + \left(47 a + 58\right)\cdot 97^{2} + \left(63 a + 67\right)\cdot 97^{3} + \left(96 a + 4\right)\cdot 97^{4} + \left(68 a + 55\right)\cdot 97^{5} + \left(21 a + 23\right)\cdot 97^{6} + \left(94 a + 61\right)\cdot 97^{7} + \left(76 a + 13\right)\cdot 97^{8} + \left(38 a + 53\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 32 a + 16 + \left(56 a + 89\right)\cdot 97 + \left(5 a + 17\right)\cdot 97^{2} + \left(38 a + 41\right)\cdot 97^{3} + \left(40 a + 23\right)\cdot 97^{4} + \left(43 a + 88\right)\cdot 97^{5} + \left(59 a + 53\right)\cdot 97^{6} + \left(46 a + 51\right)\cdot 97^{7} + \left(19 a + 11\right)\cdot 97^{8} + \left(35 a + 18\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 65 a + 48 + \left(40 a + 16\right)\cdot 97 + \left(91 a + 64\right)\cdot 97^{2} + \left(58 a + 73\right)\cdot 97^{3} + \left(56 a + 25\right)\cdot 97^{4} + \left(53 a + 91\right)\cdot 97^{5} + \left(37 a + 69\right)\cdot 97^{6} + \left(50 a + 38\right)\cdot 97^{7} + \left(77 a + 81\right)\cdot 97^{8} + \left(61 a + 33\right)\cdot 97^{9} +O(97^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,4)(3,6)$ | $-6$ |
| $9$ | $2$ | $(1,7)(2,4)(3,6)(5,8)$ | $2$ |
| $12$ | $2$ | $(2,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $36$ | $2$ | $(1,5)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,7)(2,3)(5,8)$ | $0$ |
| $16$ | $3$ | $(3,4,6)$ | $0$ |
| $64$ | $3$ | $(2,4,6)(5,7,8)$ | $0$ |
| $12$ | $4$ | $(2,3,4,6)$ | $0$ |
| $36$ | $4$ | $(1,5,7,8)(2,3,4,6)$ | $-2$ |
| $36$ | $4$ | $(1,5,7,8)(2,4)(3,6)$ | $0$ |
| $72$ | $4$ | $(1,2,7,4)(3,8,6,5)$ | $0$ |
| $72$ | $4$ | $(1,5,7,8)(2,3)$ | $2$ |
| $144$ | $4$ | $(1,2,5,3)(4,7)(6,8)$ | $0$ |
| $48$ | $6$ | $(1,7)(3,6,4)(5,8)$ | $0$ |
| $96$ | $6$ | $(2,3)(5,8,7)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,5,4,7,6,8)$ | $0$ |
| $144$ | $8$ | $(1,2,5,3,7,4,8,6)$ | $0$ |
| $96$ | $12$ | $(1,5,7,8)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.