Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(27551316672\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 11^{6} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.164627620608.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T311 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.164627620608.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} - 2x^{7} + 5x^{6} - 10x^{5} + 6x^{4} + 24x^{3} - 20x^{2} - 8x + 24 \)
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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 $\mathbb{Q}_{ 97 }$:
\( x^{3} + 9x + 92 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 91 a^{2} + 87 a + 56 + \left(87 a^{2} + 55 a + 75\right)\cdot 97 + \left(45 a^{2} + 53 a + 64\right)\cdot 97^{2} + \left(17 a^{2} + 61 a + 28\right)\cdot 97^{3} + \left(82 a^{2} + 16 a + 90\right)\cdot 97^{4} + \left(76 a^{2} + 9 a + 4\right)\cdot 97^{5} + \left(55 a^{2} + 43 a + 58\right)\cdot 97^{6} + \left(12 a^{2} + 65 a + 88\right)\cdot 97^{7} + \left(39 a^{2} + 84 a + 72\right)\cdot 97^{8} + \left(95 a^{2} + 4 a + 25\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 44 a^{2} + 87 a + 72 + \left(56 a^{2} + 94 a + 37\right)\cdot 97 + \left(39 a^{2} + 89 a + 27\right)\cdot 97^{2} + \left(87 a^{2} + 39 a + 9\right)\cdot 97^{3} + \left(89 a^{2} + 40 a + 49\right)\cdot 97^{4} + \left(35 a^{2} + 79 a + 56\right)\cdot 97^{5} + \left(32 a^{2} + 42 a + 61\right)\cdot 97^{6} + \left(56 a^{2} + 76 a + 20\right)\cdot 97^{7} + \left(65 a^{2} + 83 a + 67\right)\cdot 97^{8} + \left(86 a^{2} + 95 a + 68\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a^{2} + 93 a + 13 + \left(94 a^{2} + 51 a + 70\right)\cdot 97 + \left(78 a^{2} + 90 a + 69\right)\cdot 97^{2} + \left(37 a^{2} + 76 a + 2\right)\cdot 97^{3} + \left(29 a^{2} + 59 a + 74\right)\cdot 97^{4} + \left(82 a^{2} + 65 a + 43\right)\cdot 97^{5} + \left(78 a^{2} + 83 a + 49\right)\cdot 97^{6} + \left(86 a^{2} + 22 a + 9\right)\cdot 97^{7} + \left(34 a^{2} + 7 a + 77\right)\cdot 97^{8} + \left(81 a^{2} + 63 a + 36\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 69 a^{2} + 43 a + 94 + \left(13 a^{2} + 17 a + 26\right)\cdot 97 + \left(22 a^{2} + 85 a + 36\right)\cdot 97^{2} + \left(45 a^{2} + 48 a + 54\right)\cdot 97^{3} + \left(81 a^{2} + 70 a + 88\right)\cdot 97^{4} + \left(12 a^{2} + 32 a + 77\right)\cdot 97^{5} + \left(39 a^{2} + 22 a + 29\right)\cdot 97^{6} + \left(86 a^{2} + 77 a + 14\right)\cdot 97^{7} + \left(44 a^{2} + 84 a + 46\right)\cdot 97^{8} + \left(5 a^{2} + 81 a + 28\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 35 a^{2} + 83 a + 11 + \left(49 a^{2} + 79 a + 38\right)\cdot 97 + \left(47 a^{2} + 72 a + 74\right)\cdot 97^{2} + \left(18 a^{2} + 23 a + 34\right)\cdot 97^{3} + \left(27 a^{2} + 64 a + 51\right)\cdot 97^{4} + \left(11 a^{2} + 8 a + 96\right)\cdot 97^{5} + \left(55 a^{2} + 60 a + 53\right)\cdot 97^{6} + \left(78 a^{2} + 53 a + 96\right)\cdot 97^{7} + \left(48 a^{2} + 77 a + 33\right)\cdot 97^{8} + \left(25 a^{2} + 77 a + 91\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 68 a^{2} + 96 a + 88 + \left(7 a^{2} + 35 a + 87\right)\cdot 97 + \left(72 a^{2} + 52 a + 44\right)\cdot 97^{2} + \left(36 a^{2} + 47 a + 3\right)\cdot 97^{3} + \left(23 a^{2} + 9 a + 31\right)\cdot 97^{4} + \left(69 a^{2} + 30 a + 28\right)\cdot 97^{5} + \left(26 a^{2} + 68 a + 52\right)\cdot 97^{6} + \left(34 a^{2} + 77 a + 89\right)\cdot 97^{7} + \left(45 a^{2} + 70 a + 48\right)\cdot 97^{8} + \left(44 a^{2} + 82 a + 68\right)\cdot 97^{9} +O(97^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 35 a^{2} + 14 a + 18 + \left(43 a^{2} + 47 a + 56\right)\cdot 97 + \left(75 a^{2} + 13 a + 48\right)\cdot 97^{2} + \left(68 a^{2} + 77 a + 91\right)\cdot 97^{3} + \left(74 a^{2} + 93 a + 54\right)\cdot 97^{4} + \left(75 a^{2} + 48 a + 4\right)\cdot 97^{5} + \left(82 a^{2} + 67 a + 73\right)\cdot 97^{6} + \left(50 a^{2} + 94 a + 84\right)\cdot 97^{7} + \left(93 a^{2} + 5 a + 40\right)\cdot 97^{8} + \left(25 a^{2} + 35 a + 92\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 68 a^{2} + 24 a + 15 + \left(56 a^{2} + 58 a + 82\right)\cdot 97 + \left(3 a^{2} + 67 a + 4\right)\cdot 97^{2} + \left(61 a^{2} + 11 a + 96\right)\cdot 97^{3} + \left(84 a^{2} + 16 a + 7\right)\cdot 97^{4} + \left(8 a^{2} + 79 a + 82\right)\cdot 97^{5} + \left(83 a^{2} + 90 a + 27\right)\cdot 97^{6} + \left(5 a^{2} + 74 a + 48\right)\cdot 97^{7} + \left(9 a^{2} + 31 a + 86\right)\cdot 97^{8} + \left(73 a^{2} + 14 a + 85\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 57 a^{2} + 55 a + 22 + \left(75 a^{2} + 43 a + 10\right)\cdot 97 + \left(2 a^{2} + 56 a + 17\right)\cdot 97^{2} + \left(15 a^{2} + 67\right)\cdot 97^{3} + \left(89 a^{2} + 17 a + 37\right)\cdot 97^{4} + \left(14 a^{2} + 34 a + 90\right)\cdot 97^{5} + \left(31 a^{2} + 6 a + 78\right)\cdot 97^{6} + \left(73 a^{2} + 39 a + 32\right)\cdot 97^{7} + \left(6 a^{2} + 38 a + 11\right)\cdot 97^{8} + \left(47 a^{2} + 29 a + 84\right)\cdot 97^{9} +O(97^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $6$ | |
| $9$ | $2$ | $(1,2)$ | $-4$ | |
| $18$ | $2$ | $(1,3)(2,4)(5,9)$ | $2$ | |
| $27$ | $2$ | $(1,2)(3,4)(6,7)$ | $0$ | |
| $27$ | $2$ | $(1,2)(3,4)$ | $2$ | |
| $54$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $0$ | ✓ |
| $6$ | $3$ | $(6,7,8)$ | $3$ | |
| $8$ | $3$ | $(1,2,9)(3,4,5)(6,7,8)$ | $-3$ | |
| $12$ | $3$ | $(3,4,5)(6,7,8)$ | $0$ | |
| $72$ | $3$ | $(1,3,6)(2,4,7)(5,8,9)$ | $0$ | |
| $54$ | $4$ | $(1,4,2,3)(5,9)$ | $-2$ | |
| $162$ | $4$ | $(1,7,2,6)(4,5)(8,9)$ | $0$ | |
| $36$ | $6$ | $(1,3)(2,4)(5,9)(6,7,8)$ | $-1$ | |
| $36$ | $6$ | $(1,6,2,7,9,8)$ | $2$ | |
| $36$ | $6$ | $(1,2)(6,7,8)$ | $-1$ | |
| $36$ | $6$ | $(1,2)(3,4,5)(6,7,8)$ | $2$ | |
| $54$ | $6$ | $(1,2)(3,4)(6,8,7)$ | $-1$ | |
| $72$ | $6$ | $(1,3,2,4,9,5)(6,7,8)$ | $-1$ | |
| $108$ | $6$ | $(1,2)(3,6,4,7,5,8)$ | $0$ | |
| $216$ | $6$ | $(1,4,7,2,3,6)(5,8,9)$ | $0$ | |
| $144$ | $9$ | $(1,3,6,2,4,7,9,5,8)$ | $0$ | |
| $108$ | $12$ | $(1,4,2,3)(5,9)(6,7,8)$ | $1$ |