Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(604160000\)\(\medspace = 2^{14} \cdot 5^{4} \cdot 59 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.7552000000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T319 |
| Parity: | odd |
| Determinant: | 1.59.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.7552000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} + 3x^{7} - 2x^{6} + 3x^{5} - 4x^{4} - x^{3} - 2x^{2} + 8x - 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{3} + x + 35 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a + 29 + \left(18 a^{2} + 19 a + 9\right)\cdot 41 + \left(16 a^{2} + 30 a + 22\right)\cdot 41^{2} + \left(8 a^{2} + 22 a + 9\right)\cdot 41^{3} + \left(4 a^{2} + 7 a + 6\right)\cdot 41^{4} + \left(7 a^{2} + 33 a + 13\right)\cdot 41^{5} + \left(20 a^{2} + 13 a + 11\right)\cdot 41^{6} + \left(39 a^{2} + 26 a + 15\right)\cdot 41^{7} + \left(3 a^{2} + 3 a + 13\right)\cdot 41^{8} + \left(22 a^{2} + 19 a + 33\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 a + 34 + \left(5 a^{2} + 7 a + 29\right)\cdot 41 + \left(27 a^{2} + 14 a + 8\right)\cdot 41^{2} + \left(a^{2} + 11 a + 30\right)\cdot 41^{3} + \left(32 a^{2} + 19 a + 10\right)\cdot 41^{4} + \left(4 a^{2} + 18 a + 4\right)\cdot 41^{5} + \left(12 a^{2} + 16 a + 23\right)\cdot 41^{6} + \left(14 a^{2} + 25 a + 32\right)\cdot 41^{7} + \left(5 a^{2} + 30 a + 36\right)\cdot 41^{8} + \left(36 a^{2} + 20 a + 17\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a^{2} + 12 a + 21 + \left(20 a^{2} + 33 a + 16\right)\cdot 41 + \left(39 a^{2} + 8 a + 24\right)\cdot 41^{2} + \left(10 a^{2} + 25 a + 1\right)\cdot 41^{3} + \left(14 a^{2} + 30 a + 3\right)\cdot 41^{4} + \left(14 a^{2} + 24 a\right)\cdot 41^{5} + \left(2 a^{2} + 26 a + 16\right)\cdot 41^{6} + \left(3 a^{2} + 23 a + 17\right)\cdot 41^{7} + \left(20 a^{2} + 36 a + 10\right)\cdot 41^{8} + \left(28 a^{2} + 17 a + 20\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 a^{2} + 34 a + 2 + \left(37 a^{2} + 22 a + 38\right)\cdot 41 + \left(36 a^{2} + 17 a + 28\right)\cdot 41^{2} + \left(22 a^{2} + 37 a + 30\right)\cdot 41^{3} + \left(19 a^{2} + 31 a + 29\right)\cdot 41^{4} + \left(40 a + 28\right)\cdot 41^{5} + \left(13 a^{2} + a + 23\right)\cdot 41^{6} + \left(18 a^{2} + 37 a + 21\right)\cdot 41^{7} + \left(5 a^{2} + a + 9\right)\cdot 41^{8} + \left(35 a^{2} + 26 a + 17\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 32 a^{2} + 11 a + 23 + \left(39 a^{2} + 23 a + 10\right)\cdot 41 + \left(4 a^{2} + 36 a + 28\right)\cdot 41^{2} + \left(13 a^{2} + 9 a + 12\right)\cdot 41^{3} + \left(12 a^{2} + 34 a + 25\right)\cdot 41^{4} + \left(25 a^{2} + a + 11\right)\cdot 41^{5} + \left(9 a^{2} + 14 a + 4\right)\cdot 41^{6} + \left(20 a^{2} + 32 a + 16\right)\cdot 41^{7} + \left(24 a^{2} + 22 a + 13\right)\cdot 41^{8} + \left(16 a^{2} + 3 a + 2\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 a^{2} + 29 a + 2 + \left(22 a^{2} + 17 a + 32\right)\cdot 41 + \left(11 a^{2} + 24 a + 5\right)\cdot 41^{2} + \left(31 a + 8\right)\cdot 41^{3} + \left(34 a^{2} + 26 a + 16\right)\cdot 41^{4} + \left(14 a^{2} + 28 a\right)\cdot 41^{5} + \left(35 a^{2} + 21 a + 38\right)\cdot 41^{6} + \left(30 a^{2} + 29 a + 35\right)\cdot 41^{7} + \left(17 a^{2} + 18 a + 8\right)\cdot 41^{8} + \left(13 a^{2} + 14 a + 10\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 2 a^{2} + 34 + \left(39 a^{2} + 31 a + 1\right)\cdot 41 + \left(30 a^{2} + 7 a + 5\right)\cdot 41^{2} + \left(29 a^{2} + 25 a + 14\right)\cdot 41^{3} + \left(33 a^{2} + 24 a + 2\right)\cdot 41^{4} + \left(11 a^{2} + 28 a + 12\right)\cdot 41^{5} + \left(3 a^{2} + 33 a + 30\right)\cdot 41^{6} + \left(7 a^{2} + 28 a + 33\right)\cdot 41^{7} + \left(3 a^{2} + 26 a + 12\right)\cdot 41^{8} + \left(40 a^{2} + 8 a + 14\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 9 a^{2} + 9 a + 35 + \left(24 a^{2} + 39 a + 13\right)\cdot 41 + \left(19 a^{2} + 14 a + 24\right)\cdot 41^{2} + \left(19 a^{2} + 8 a + 30\right)\cdot 41^{3} + \left(24 a^{2} + 40 a + 19\right)\cdot 41^{4} + \left(8 a^{2} + 5 a\right)\cdot 41^{5} + \left(11 a^{2} + 13 a + 19\right)\cdot 41^{6} + \left(22 a^{2} + 23 a + 17\right)\cdot 41^{7} + \left(12 a^{2} + 14 a + 5\right)\cdot 41^{8} + \left(2 a^{2} + 18 a + 20\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 7 a^{2} + 37 a + 25 + \left(39 a^{2} + 10 a + 11\right)\cdot 41 + \left(17 a^{2} + 9 a + 16\right)\cdot 41^{2} + \left(16 a^{2} + 33 a + 26\right)\cdot 41^{3} + \left(30 a^{2} + 30 a + 9\right)\cdot 41^{4} + \left(35 a^{2} + 22 a + 11\right)\cdot 41^{5} + \left(15 a^{2} + 22 a + 39\right)\cdot 41^{6} + \left(8 a^{2} + 19 a + 14\right)\cdot 41^{7} + \left(30 a^{2} + 8 a + 12\right)\cdot 41^{8} + \left(10 a^{2} + 35 a + 28\right)\cdot 41^{9} +O(41^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)$ | $4$ |
| $18$ | $2$ | $(1,3)(2,5)(6,9)$ | $-2$ |
| $27$ | $2$ | $(1,2)(3,5)$ | $2$ |
| $27$ | $2$ | $(1,2)(3,5)(4,7)$ | $0$ |
| $54$ | $2$ | $(1,4)(2,7)(3,5)(6,8)$ | $0$ |
| $6$ | $3$ | $(4,7,8)$ | $3$ |
| $8$ | $3$ | $(1,2,6)(3,5,9)(4,7,8)$ | $-3$ |
| $12$ | $3$ | $(1,2,6)(4,7,8)$ | $0$ |
| $72$ | $3$ | $(1,3,4)(2,5,7)(6,9,8)$ | $0$ |
| $54$ | $4$ | $(1,3,2,5)(6,9)$ | $-2$ |
| $162$ | $4$ | $(1,7,2,4)(3,5)(6,8)$ | $0$ |
| $36$ | $6$ | $(1,3)(2,5)(4,7,8)(6,9)$ | $1$ |
| $36$ | $6$ | $(1,4,2,7,6,8)$ | $-2$ |
| $36$ | $6$ | $(1,2)(4,7,8)$ | $1$ |
| $36$ | $6$ | $(1,2)(3,5,9)(4,7,8)$ | $-2$ |
| $54$ | $6$ | $(1,2)(3,5)(4,8,7)$ | $-1$ |
| $72$ | $6$ | $(1,3,6,9,2,5)(4,7,8)$ | $1$ |
| $108$ | $6$ | $(1,4,2,7,6,8)(3,5)$ | $0$ |
| $216$ | $6$ | $(1,5,7,2,3,4)(6,9,8)$ | $0$ |
| $144$ | $9$ | $(1,3,4,2,5,7,6,9,8)$ | $0$ |
| $108$ | $12$ | $(1,3,2,5)(4,7,8)(6,9)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.