Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(15479341056\)\(\medspace = 2^{18} \cdot 3^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.13060694016.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.13060694016.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{6} + 3x^{5} - 3x^{4} - 6x^{2} + 6x - 2 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$:
\( x^{3} + 3x + 99 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 66 a^{2} + 38 a + 91 + \left(20 a^{2} + 72 a + 42\right)\cdot 101 + \left(31 a^{2} + 100 a + 23\right)\cdot 101^{2} + \left(38 a^{2} + 31 a + 21\right)\cdot 101^{3} + \left(41 a^{2} + 69 a + 14\right)\cdot 101^{4} + \left(22 a^{2} + 54 a + 70\right)\cdot 101^{5} + \left(58 a^{2} + 48 a + 9\right)\cdot 101^{6} + \left(81 a^{2} + 84 a + 88\right)\cdot 101^{7} + \left(58 a^{2} + 57 a + 54\right)\cdot 101^{8} + \left(84 a^{2} + 49 a + 34\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 95 a^{2} + 19 a + 26 + \left(99 a^{2} + 75 a + 22\right)\cdot 101 + \left(6 a^{2} + 21 a + 57\right)\cdot 101^{2} + \left(55 a^{2} + 97 a + 14\right)\cdot 101^{3} + \left(29 a + 29\right)\cdot 101^{4} + \left(37 a^{2} + 53 a + 84\right)\cdot 101^{5} + \left(75 a^{2} + 20 a + 66\right)\cdot 101^{6} + \left(74 a^{2} + 27 a + 54\right)\cdot 101^{7} + \left(15 a^{2} + 17 a + 51\right)\cdot 101^{8} + \left(26 a^{2} + 32 a + 87\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 71 a^{2} + 42 a + \left(9 a^{2} + 88 a + 21\right)\cdot 101 + \left(92 a^{2} + 80 a + 44\right)\cdot 101^{2} + \left(72 a^{2} + 95 a + 90\right)\cdot 101^{3} + \left(88 a^{2} + 15 a + 7\right)\cdot 101^{4} + \left(46 a^{2} + 25 a + 18\right)\cdot 101^{5} + \left(84 a^{2} + 30 a + 62\right)\cdot 101^{6} + \left(50 a^{2} + 30 a + 26\right)\cdot 101^{7} + \left(96 a^{2} + 96 a + 29\right)\cdot 101^{8} + \left(23 a^{2} + 31 a + 14\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 40 a^{2} + 36 a + 17 + \left(11 a^{2} + a + 47\right)\cdot 101 + \left(11 a^{2} + 36 a + 65\right)\cdot 101^{2} + \left(49 a^{2} + 82 a + 2\right)\cdot 101^{3} + \left(69 a^{2} + 82 a + 66\right)\cdot 101^{4} + \left(88 a^{2} + 79 a + 86\right)\cdot 101^{5} + \left(30 a^{2} + 90 a + 78\right)\cdot 101^{6} + \left(53 a^{2} + 27 a + 11\right)\cdot 101^{7} + \left(96 a^{2} + 88 a + 11\right)\cdot 101^{8} + \left(23 a^{2} + 28 a + 83\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 66 a^{2} + 88 a + 34 + \left(18 a^{2} + 15 a + 11\right)\cdot 101 + \left(55 a^{2} + 6 a + 5\right)\cdot 101^{2} + \left(88 a^{2} + 99 a + 25\right)\cdot 101^{3} + \left(48 a^{2} + 51 a + 37\right)\cdot 101^{4} + \left(36 a^{2} + 30 a + 37\right)\cdot 101^{5} + \left(11 a^{2} + 46 a + 11\right)\cdot 101^{6} + \left(2 a^{2} + 58 a + 73\right)\cdot 101^{7} + \left(52 a^{2} + 6 a + 45\right)\cdot 101^{8} + \left(73 a^{2} + 73 a + 44\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 65 a^{2} + 21 a + 89 + \left(70 a^{2} + 41 a + 41\right)\cdot 101 + \left(78 a^{2} + 20 a + 17\right)\cdot 101^{2} + \left(90 a^{2} + 74 a + 25\right)\cdot 101^{3} + \left(71 a^{2} + 15 a + 75\right)\cdot 101^{4} + \left(31 a^{2} + 21 a + 88\right)\cdot 101^{5} + \left(59 a^{2} + 22 a + 11\right)\cdot 101^{6} + \left(69 a^{2} + 87 a + 64\right)\cdot 101^{7} + \left(46 a^{2} + 47 a + 30\right)\cdot 101^{8} + \left(93 a^{2} + 19 a + 52\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 67 a^{2} + 46 a + 71 + \left(90 a^{2} + 24 a + 3\right)\cdot 101 + \left(82 a^{2} + 43 a + 7\right)\cdot 101^{2} + \left(97 a^{2} + 22 a + 100\right)\cdot 101^{3} + \left(30 a^{2} + 89 a + 89\right)\cdot 101^{4} + \left(76 a^{2} + 68 a + 61\right)\cdot 101^{5} + \left(95 a^{2} + 90 a + 6\right)\cdot 101^{6} + \left(73 a^{2} + 45 a + 53\right)\cdot 101^{7} + \left(89 a^{2} + 96 a + 98\right)\cdot 101^{8} + \left(50 a^{2} + 39 a + 35\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 44 a^{2} + 91 a + 91 + \left(86 a^{2} + 90 a + 45\right)\cdot 101 + \left(17 a^{2} + 11 a + 31\right)\cdot 101^{2} + \left(2 a^{2} + 84 a + 54\right)\cdot 101^{3} + \left(53 a^{2} + 40 a + 45\right)\cdot 101^{4} + \left(60 a^{2} + 100 a + 85\right)\cdot 101^{5} + \left(8 a^{2} + 46 a + 5\right)\cdot 101^{6} + \left(42 a^{2} + 72 a + 52\right)\cdot 101^{7} + \left(23 a^{2} + 79 a + 89\right)\cdot 101^{8} + \left(86 a^{2} + 79 a + 69\right)\cdot 101^{9} +O(101^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 92 a^{2} + 23 a + 86 + \left(96 a^{2} + 95 a + 66\right)\cdot 101 + \left(27 a^{2} + 82 a + 51\right)\cdot 101^{2} + \left(10 a^{2} + 18 a + 70\right)\cdot 101^{3} + \left(100 a^{2} + 8 a + 38\right)\cdot 101^{4} + \left(3 a^{2} + 71 a + 73\right)\cdot 101^{5} + \left(81 a^{2} + 7 a + 49\right)\cdot 101^{6} + \left(56 a^{2} + 71 a + 81\right)\cdot 101^{7} + \left(25 a^{2} + 14 a + 93\right)\cdot 101^{8} + \left(42 a^{2} + 49 a + 82\right)\cdot 101^{9} +O(101^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(5,6)$ | $0$ |
| $18$ | $2$ | $(1,5)(2,6)(7,8)$ | $4$ |
| $27$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $27$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,4)(5,6)(8,9)$ | $0$ |
| $6$ | $3$ | $(3,4,9)$ | $-4$ |
| $8$ | $3$ | $(1,2,8)(3,4,9)(5,6,7)$ | $-1$ |
| $12$ | $3$ | $(1,2,8)(3,4,9)$ | $2$ |
| $72$ | $3$ | $(1,3,5)(2,4,6)(7,8,9)$ | $2$ |
| $54$ | $4$ | $(1,5,2,6)(7,8)$ | $0$ |
| $162$ | $4$ | $(2,8)(3,5,4,6)(7,9)$ | $0$ |
| $36$ | $6$ | $(1,5)(2,6)(3,4,9)(7,8)$ | $-2$ |
| $36$ | $6$ | $(3,6,4,7,9,5)$ | $-2$ |
| $36$ | $6$ | $(3,4,9)(5,6)$ | $0$ |
| $36$ | $6$ | $(1,2,8)(3,4,9)(5,6)$ | $0$ |
| $54$ | $6$ | $(1,2)(3,9,4)(5,6)$ | $0$ |
| $72$ | $6$ | $(1,6,2,7,8,5)(3,4,9)$ | $1$ |
| $108$ | $6$ | $(1,3,2,4,8,9)(5,6)$ | $0$ |
| $216$ | $6$ | $(1,3,5,2,4,6)(7,8,9)$ | $0$ |
| $144$ | $9$ | $(1,3,6,2,4,7,8,9,5)$ | $-1$ |
| $108$ | $12$ | $(1,5,2,6)(3,4,9)(7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.