Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(16912482304\)\(\medspace = 2^{20} \cdot 127^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.3.67121414144.1 |
| 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 9.3.67121414144.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 4x^{8} + 9x^{7} - 6x^{6} - 7x^{5} + 12x^{4} + 5x^{3} - 2x^{2} - 8x - 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 24 + 17\cdot 29 + 17\cdot 29^{2} + 5\cdot 29^{3} + 16\cdot 29^{4} + 19\cdot 29^{5} + 25\cdot 29^{6} + 21\cdot 29^{7} + 26\cdot 29^{8} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 + 25\cdot 29 + 17\cdot 29^{2} + 8\cdot 29^{4} + 4\cdot 29^{6} + 19\cdot 29^{7} + 19\cdot 29^{8} + 3\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a^{2} + 20 a + 21 + \left(27 a^{2} + a + 9\right)\cdot 29 + \left(13 a^{2} + 10 a + 27\right)\cdot 29^{2} + \left(23 a^{2} + 15 a + 28\right)\cdot 29^{3} + \left(24 a^{2} + 18 a + 15\right)\cdot 29^{4} + \left(4 a^{2} + 23 a + 22\right)\cdot 29^{5} + \left(19 a^{2} + 26 a + 2\right)\cdot 29^{6} + \left(5 a^{2} + 5 a\right)\cdot 29^{7} + \left(16 a^{2} + 21 a + 15\right)\cdot 29^{8} + \left(a^{2} + 10 a + 8\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a^{2} + 7 a + 25 + \left(a^{2} + 2 a + 13\right)\cdot 29 + \left(27 a^{2} + 6 a + 23\right)\cdot 29^{2} + \left(23 a^{2} + 8 a + 16\right)\cdot 29^{3} + \left(22 a^{2} + 26 a + 5\right)\cdot 29^{4} + \left(21 a^{2} + 23 a + 12\right)\cdot 29^{5} + \left(11 a^{2} + 13 a + 6\right)\cdot 29^{6} + \left(21 a^{2} + 2 a + 5\right)\cdot 29^{7} + \left(20 a^{2} + 3 a + 10\right)\cdot 29^{8} + \left(22 a^{2} + a + 14\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 3 a^{2} + 21 a + 10 + \left(20 a^{2} + 2 a\right)\cdot 29 + \left(11 a^{2} + 16 a + 5\right)\cdot 29^{2} + \left(3 a^{2} + 19 a + 2\right)\cdot 29^{3} + \left(27 a^{2} + 22 a + 19\right)\cdot 29^{4} + \left(27 a^{2} + 25 a + 14\right)\cdot 29^{5} + \left(7 a^{2} + 24 a + 26\right)\cdot 29^{6} + \left(19 a^{2} + 11 a + 27\right)\cdot 29^{7} + \left(17 a^{2} + 7 a + 16\right)\cdot 29^{8} + \left(7 a^{2} + 6 a + 16\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a^{2} + 3 a + 11 + \left(3 a^{2} + 20 a + 26\right)\cdot 29 + \left(5 a^{2} + 11 a + 3\right)\cdot 29^{2} + \left(2 a^{2} + 26 a + 7\right)\cdot 29^{3} + \left(14 a^{2} + 16 a + 13\right)\cdot 29^{4} + \left(7 a^{2} + 26 a + 12\right)\cdot 29^{5} + \left(14 a^{2} + 18 a + 19\right)\cdot 29^{6} + \left(24 a^{2} + 20 a + 28\right)\cdot 29^{7} + \left(19 a^{2} + 13 a + 8\right)\cdot 29^{8} + \left(16 a^{2} + 2 a + 6\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 22 a^{2} + 17 a + 16 + \left(10 a^{2} + 24 a + 26\right)\cdot 29 + \left(3 a^{2} + 2 a + 22\right)\cdot 29^{2} + \left(2 a^{2} + 23 a + 19\right)\cdot 29^{3} + \left(6 a^{2} + 16 a\right)\cdot 29^{4} + \left(25 a^{2} + 8 a + 11\right)\cdot 29^{5} + \left(a^{2} + 6 a + 18\right)\cdot 29^{6} + \left(4 a^{2} + 11 a + 7\right)\cdot 29^{7} + \left(24 a^{2} + 6\right)\cdot 29^{8} + \left(19 a^{2} + 12 a + 23\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 17 + 5\cdot 29^{2} + 17\cdot 29^{3} + 14\cdot 29^{4} + 11\cdot 29^{5} + 8\cdot 29^{6} + 22\cdot 29^{7} + 25\cdot 29^{8} + 23\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 28 a^{2} + 19 a + 16 + \left(23 a^{2} + 6 a + 24\right)\cdot 29 + \left(25 a^{2} + 11 a + 21\right)\cdot 29^{2} + \left(2 a^{2} + 23 a + 17\right)\cdot 29^{3} + \left(21 a^{2} + 14 a + 22\right)\cdot 29^{4} + \left(28 a^{2} + 7 a + 11\right)\cdot 29^{5} + \left(2 a^{2} + 25 a + 4\right)\cdot 29^{6} + \left(12 a^{2} + 5 a + 12\right)\cdot 29^{7} + \left(17 a^{2} + 12 a + 15\right)\cdot 29^{8} + \left(18 a^{2} + 25 a + 18\right)\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,2)$ | $0$ |
| $18$ | $2$ | $(1,4)(2,6)(8,9)$ | $4$ |
| $27$ | $2$ | $(1,2)(4,6)$ | $0$ |
| $27$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $0$ |
| $6$ | $3$ | $(3,5,7)$ | $-4$ |
| $8$ | $3$ | $(1,2,8)(3,5,7)(4,6,9)$ | $-1$ |
| $12$ | $3$ | $(1,2,8)(3,5,7)$ | $2$ |
| $72$ | $3$ | $(1,4,3)(2,6,5)(7,8,9)$ | $2$ |
| $54$ | $4$ | $(1,4,2,6)(8,9)$ | $0$ |
| $162$ | $4$ | $(1,5,2,3)(4,6)(7,8)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,6)(3,5,7)(8,9)$ | $-2$ |
| $36$ | $6$ | $(1,3,2,5,8,7)$ | $-2$ |
| $36$ | $6$ | $(1,2)(3,5,7)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,5,7)(4,6,9)$ | $0$ |
| $54$ | $6$ | $(1,2)(3,7,5)(4,6)$ | $0$ |
| $72$ | $6$ | $(1,4,8,9,2,6)(3,5,7)$ | $1$ |
| $108$ | $6$ | $(1,3,2,5,8,7)(4,6)$ | $0$ |
| $216$ | $6$ | $(1,6,5,2,4,3)(7,8,9)$ | $0$ |
| $144$ | $9$ | $(1,4,3,2,6,5,8,9,7)$ | $-1$ |
| $108$ | $12$ | $(1,4,2,6)(3,5,7)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.