Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(129600\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1119744000.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8:S_4$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2:S_4$ |
| Projective stem field: | Galois closure of 8.0.6561000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 12x^{6} - 22x^{5} + 37x^{4} - 42x^{3} + 40x^{2} - 22x + 3 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 72 a + 13 + \left(31 a + 20\right)\cdot 97 + \left(64 a + 32\right)\cdot 97^{2} + \left(16 a + 72\right)\cdot 97^{3} + \left(59 a + 75\right)\cdot 97^{4} + \left(28 a + 63\right)\cdot 97^{5} + \left(82 a + 21\right)\cdot 97^{6} + \left(87 a + 94\right)\cdot 97^{7} + \left(90 a + 46\right)\cdot 97^{8} + \left(12 a + 87\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 33 a + 43 + \left(35 a + 78\right)\cdot 97 + \left(5 a + 8\right)\cdot 97^{2} + \left(2 a + 7\right)\cdot 97^{3} + \left(71 a + 32\right)\cdot 97^{4} + \left(85 a + 64\right)\cdot 97^{5} + \left(32 a + 85\right)\cdot 97^{6} + \left(69 a + 70\right)\cdot 97^{7} + \left(45 a + 25\right)\cdot 97^{8} + \left(29 a + 3\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 33 a + 22 + \left(35 a + 16\right)\cdot 97 + \left(5 a + 21\right)\cdot 97^{2} + \left(2 a + 93\right)\cdot 97^{3} + \left(71 a + 92\right)\cdot 97^{4} + \left(85 a + 17\right)\cdot 97^{5} + \left(32 a + 64\right)\cdot 97^{6} + \left(69 a + 86\right)\cdot 97^{7} + \left(45 a + 94\right)\cdot 97^{8} + \left(29 a + 12\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 + 47\cdot 97 + 60\cdot 97^{2} + 28\cdot 97^{3} + 43\cdot 97^{4} + 58\cdot 97^{5} + 31\cdot 97^{6} + 61\cdot 97^{7} + 71\cdot 97^{8} + 32\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 a + 76 + \left(61 a + 80\right)\cdot 97 + \left(91 a + 75\right)\cdot 97^{2} + \left(94 a + 3\right)\cdot 97^{3} + \left(25 a + 4\right)\cdot 97^{4} + \left(11 a + 79\right)\cdot 97^{5} + \left(64 a + 32\right)\cdot 97^{6} + \left(27 a + 10\right)\cdot 97^{7} + \left(51 a + 2\right)\cdot 97^{8} + \left(67 a + 84\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 a + 55 + \left(61 a + 18\right)\cdot 97 + \left(91 a + 88\right)\cdot 97^{2} + \left(94 a + 89\right)\cdot 97^{3} + \left(25 a + 64\right)\cdot 97^{4} + \left(11 a + 32\right)\cdot 97^{5} + \left(64 a + 11\right)\cdot 97^{6} + \left(27 a + 26\right)\cdot 97^{7} + \left(51 a + 71\right)\cdot 97^{8} + \left(67 a + 93\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 91 + 49\cdot 97 + 36\cdot 97^{2} + 68\cdot 97^{3} + 53\cdot 97^{4} + 38\cdot 97^{5} + 65\cdot 97^{6} + 35\cdot 97^{7} + 25\cdot 97^{8} + 64\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 25 a + 85 + \left(65 a + 76\right)\cdot 97 + \left(32 a + 64\right)\cdot 97^{2} + \left(80 a + 24\right)\cdot 97^{3} + \left(37 a + 21\right)\cdot 97^{4} + \left(68 a + 33\right)\cdot 97^{5} + \left(14 a + 75\right)\cdot 97^{6} + \left(9 a + 2\right)\cdot 97^{7} + \left(6 a + 50\right)\cdot 97^{8} + \left(84 a + 9\right)\cdot 97^{9} +O(97^{10})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $1$ | $2$ | $(1,8)(2,6)(3,5)(4,7)$ | $-4$ | |
| $6$ | $2$ | $(3,5)(4,7)$ | $0$ | |
| $12$ | $2$ | $(1,4)(2,5)(3,6)(7,8)$ | $0$ | |
| $24$ | $2$ | $(1,6)(2,8)(4,7)$ | $0$ | ✓ |
| $32$ | $3$ | $(1,6,3)(2,5,8)$ | $1$ | |
| $6$ | $4$ | $(1,2,8,6)(3,4,5,7)$ | $0$ | |
| $6$ | $4$ | $(1,7,8,4)(2,3,6,5)$ | $0$ | |
| $12$ | $4$ | $(1,2,8,6)$ | $2$ | |
| $12$ | $4$ | $(1,2,8,6)(3,5)(4,7)$ | $-2$ | |
| $32$ | $6$ | $(1,4,3,8,7,5)(2,6)$ | $-1$ | |
| $24$ | $8$ | $(1,3,2,4,8,5,6,7)$ | $0$ | |
| $24$ | $8$ | $(1,2,7,3,8,6,4,5)$ | $0$ |