Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(31218649344\)\(\medspace = 2^{8} \cdot 3^{6} \cdot 409^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.1418714175744.1 |
| 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 9.1.1418714175744.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 10x^{6} - 10x^{5} - 8x^{4} + 28x^{3} - 14x^{2} - 19x + 21 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$:
\( x^{3} + 2x + 98 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 2 a + 31 + \left(48 a^{2} + 5 a + 37\right)\cdot 103 + \left(100 a^{2} + 28 a + 73\right)\cdot 103^{2} + \left(100 a^{2} + 10 a + 6\right)\cdot 103^{3} + \left(84 a^{2} + 19 a + 96\right)\cdot 103^{4} + \left(96 a^{2} + 32 a + 31\right)\cdot 103^{5} + \left(51 a^{2} + 50 a + 97\right)\cdot 103^{6} + \left(92 a^{2} + 77 a + 2\right)\cdot 103^{7} + \left(27 a^{2} + 60 a + 73\right)\cdot 103^{8} + \left(21 a^{2} + 21 a + 9\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 10 a^{2} + 15 a + 74 + \left(58 a^{2} + 34 a + 19\right)\cdot 103 + \left(76 a^{2} + 44 a + 7\right)\cdot 103^{2} + \left(35 a + 7\right)\cdot 103^{3} + \left(16 a^{2} + 22 a + 64\right)\cdot 103^{4} + \left(8 a^{2} + 18 a + 3\right)\cdot 103^{5} + \left(60 a^{2} + 20 a + 102\right)\cdot 103^{6} + \left(77 a^{2} + 50 a + 95\right)\cdot 103^{7} + \left(87 a^{2} + 33 a + 9\right)\cdot 103^{8} + \left(40 a^{2} + 58 a + 66\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 a^{2} + 94 a + 82 + \left(56 a^{2} + 58 a + 85\right)\cdot 103 + \left(82 a^{2} + 42 a + 83\right)\cdot 103^{2} + \left(87 a^{2} + 65 a + 88\right)\cdot 103^{3} + \left(82 a^{2} + 30 a + 15\right)\cdot 103^{4} + \left(90 a^{2} + 40 a + 45\right)\cdot 103^{5} + \left(70 a^{2} + 51 a + 13\right)\cdot 103^{6} + \left(15 a^{2} + 26 a + 82\right)\cdot 103^{7} + \left(88 a^{2} + 83 a + 44\right)\cdot 103^{8} + \left(90 a^{2} + 95 a + 98\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a^{2} + 4 a + 47 + \left(66 a^{2} + 88 a + 35\right)\cdot 103 + \left(58 a^{2} + 58 a + 96\right)\cdot 103^{2} + \left(90 a^{2} + 90 a + 70\right)\cdot 103^{3} + \left(13 a^{2} + 33 a + 61\right)\cdot 103^{4} + \left(2 a^{2} + 26 a + 38\right)\cdot 103^{5} + \left(79 a^{2} + 21 a + 55\right)\cdot 103^{6} + \left(102 a + 94\right)\cdot 103^{7} + \left(45 a^{2} + 55 a + 96\right)\cdot 103^{8} + \left(7 a^{2} + 29 a + 16\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 a^{2} + 85 a + 51 + \left(58 a^{2} + 39 a + 93\right)\cdot 103 + \left(63 a^{2} + 24 a + 102\right)\cdot 103^{2} + \left(86 a^{2} + 10 a + 99\right)\cdot 103^{3} + \left(62 a^{2} + 98 a + 23\right)\cdot 103^{4} + \left(53 a^{2} + 26 a + 4\right)\cdot 103^{5} + \left(16 a^{2} + 72 a + 75\right)\cdot 103^{6} + \left(92 a^{2} + 93 a + 44\right)\cdot 103^{7} + \left(25 a^{2} + 99 a + 71\right)\cdot 103^{8} + \left(49 a^{2} + 72\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 a^{2} + 74 a + 59 + \left(66 a^{2} + 93 a + 61\right)\cdot 103 + \left(45 a^{2} + 38 a + 34\right)\cdot 103^{2} + \left(73 a^{2} + 65 a + 4\right)\cdot 103^{3} + \left(60 a^{2} + 6 a + 98\right)\cdot 103^{4} + \left(47 a^{2} + 35 a + 34\right)\cdot 103^{5} + \left(35 a^{2} + 73 a + 75\right)\cdot 103^{6} + \left(15 a^{2} + 42 a + 71\right)\cdot 103^{7} + \left(86 a^{2} + 19 a + 47\right)\cdot 103^{8} + \left(15 a^{2} + 75 a + 2\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 64 a^{2} + 14 a + 74 + \left(81 a^{2} + 78 a + 21\right)\cdot 103 + \left(83 a^{2} + 19 a + 61\right)\cdot 103^{2} + \left(28 a^{2} + 2 a + 91\right)\cdot 103^{3} + \left(26 a^{2} + 74 a + 43\right)\cdot 103^{4} + \left(47 a^{2} + 49 a + 64\right)\cdot 103^{5} + \left(7 a^{2} + 9 a + 28\right)\cdot 103^{6} + \left(10 a^{2} + 10 a + 38\right)\cdot 103^{7} + \left(32 a^{2} + 50 a + 45\right)\cdot 103^{8} + \left(46 a^{2} + 72 a + 34\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 66 a^{2} + 27 a + 74 + \left(91 a^{2} + 4 a + 26\right)\cdot 103 + \left(59 a^{2} + 36 a + 19\right)\cdot 103^{2} + \left(31 a^{2} + 27 a + 17\right)\cdot 103^{3} + \left(60 a^{2} + 77 a + 63\right)\cdot 103^{4} + \left(61 a^{2} + 35 a + 53\right)\cdot 103^{5} + \left(15 a^{2} + 82 a + 14\right)\cdot 103^{6} + \left(98 a^{2} + 85 a + 79\right)\cdot 103^{7} + \left(91 a^{2} + 22 a + 89\right)\cdot 103^{8} + \left(65 a^{2} + 6 a + 34\right)\cdot 103^{9} +O(103^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 77 a^{2} + 97 a + 26 + \left(91 a^{2} + 9 a + 30\right)\cdot 103 + \left(46 a^{2} + 16 a + 36\right)\cdot 103^{2} + \left(14 a^{2} + 2 a + 25\right)\cdot 103^{3} + \left(4 a^{2} + 50 a + 48\right)\cdot 103^{4} + \left(4 a^{2} + 44 a + 32\right)\cdot 103^{5} + \left(75 a^{2} + 31 a + 53\right)\cdot 103^{6} + \left(9 a^{2} + 26 a + 5\right)\cdot 103^{7} + \left(30 a^{2} + 89 a + 36\right)\cdot 103^{8} + \left(74 a^{2} + 51 a + 76\right)\cdot 103^{9} +O(103^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(3,5)$ | $0$ |
| $18$ | $2$ | $(2,3)(5,6)(7,8)$ | $4$ |
| $27$ | $2$ | $(1,4)(2,6)(3,5)$ | $0$ |
| $27$ | $2$ | $(2,6)(3,5)$ | $0$ |
| $54$ | $2$ | $(1,2)(3,5)(4,6)(7,9)$ | $0$ |
| $6$ | $3$ | $(1,4,9)$ | $-4$ |
| $8$ | $3$ | $(1,4,9)(2,6,7)(3,5,8)$ | $-1$ |
| $12$ | $3$ | $(1,4,9)(2,6,7)$ | $2$ |
| $72$ | $3$ | $(1,3,2)(4,5,6)(7,9,8)$ | $2$ |
| $54$ | $4$ | $(2,3,6,5)(7,8)$ | $0$ |
| $162$ | $4$ | $(1,3,4,5)(6,7)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,4,9)(2,3)(5,6)(7,8)$ | $-2$ |
| $36$ | $6$ | $(1,5,4,8,9,3)$ | $-2$ |
| $36$ | $6$ | $(1,4,9)(3,5)$ | $0$ |
| $36$ | $6$ | $(1,4,9)(2,6,7)(3,5)$ | $0$ |
| $54$ | $6$ | $(1,9,4)(2,6)(3,5)$ | $0$ |
| $72$ | $6$ | $(1,4,9)(2,5,6,8,7,3)$ | $1$ |
| $108$ | $6$ | $(1,6,4,7,9,2)(3,5)$ | $0$ |
| $216$ | $6$ | $(1,3,6,4,5,2)(7,9,8)$ | $0$ |
| $144$ | $9$ | $(1,5,6,4,8,7,9,3,2)$ | $-1$ |
| $108$ | $12$ | $(1,4,9)(2,3,6,5)(7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.