Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(161168128\)\(\medspace = 2^{8} \cdot 11^{4} \cdot 43 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.4875335872.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T319 |
| Parity: | odd |
| Determinant: | 1.43.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.4875335872.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} + 4x^{7} - 3x^{6} + 9x^{5} - 8x^{4} + 7x^{3} - 9x^{2} + 4x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{3} + 2x + 11 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 7 a^{2} + 4 a + 12 + \left(2 a^{2} + 6 a + 11\right)\cdot 13 + \left(7 a^{2} + 2 a + 3\right)\cdot 13^{2} + \left(9 a^{2} + a + 10\right)\cdot 13^{3} + \left(9 a^{2} + 10 a\right)\cdot 13^{4} + \left(6 a^{2} + 3 a + 6\right)\cdot 13^{5} + \left(10 a^{2} + a + 10\right)\cdot 13^{6} + \left(7 a^{2} + 7 a + 6\right)\cdot 13^{7} + \left(11 a^{2} + 6 a + 7\right)\cdot 13^{8} + \left(a^{2} + 11 a + 9\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a^{2} + 6 a + 4 + \left(4 a^{2} + 2 a + 8\right)\cdot 13 + \left(4 a^{2} + 8 a + 2\right)\cdot 13^{2} + \left(11 a^{2} + 3 a + 10\right)\cdot 13^{3} + \left(8 a^{2} + 4 a + 2\right)\cdot 13^{4} + \left(a^{2} + 7 a + 2\right)\cdot 13^{5} + \left(a^{2} + 7 a + 1\right)\cdot 13^{6} + \left(12 a^{2} + 12 a + 2\right)\cdot 13^{7} + \left(8 a^{2} + 4 a + 1\right)\cdot 13^{8} + \left(3 a^{2} + 4 a + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a^{2} + 12 a + 9 + \left(5 a^{2} + 5 a + 1\right)\cdot 13 + \left(5 a^{2} + 11 a + 4\right)\cdot 13^{2} + \left(3 a^{2} + 3 a + 8\right)\cdot 13^{3} + \left(6 a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(12 a^{2} + 3 a + 12\right)\cdot 13^{5} + \left(12 a^{2} + 3\right)\cdot 13^{6} + \left(9 a^{2} + 2 a + 12\right)\cdot 13^{7} + \left(12 a^{2} + 2 a + 1\right)\cdot 13^{8} + \left(3 a^{2} + 12 a + 3\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a^{2} + a + 11 + \left(2 a^{2} + 2\right)\cdot 13 + \left(7 a^{2} + 12 a + 8\right)\cdot 13^{2} + \left(5 a^{2} + 6 a\right)\cdot 13^{3} + \left(12 a^{2} + 5 a\right)\cdot 13^{4} + \left(5 a^{2} + 11 a + 5\right)\cdot 13^{5} + \left(10 a + 1\right)\cdot 13^{6} + \left(7 a^{2} + a + 10\right)\cdot 13^{7} + \left(3 a^{2} + 9 a + 9\right)\cdot 13^{8} + \left(a^{2} + 5 a + 8\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a^{2} + 3 a + 10 + \left(6 a^{2} + 4 a + 5\right)\cdot 13 + \left(a^{2} + 2 a + 6\right)\cdot 13^{2} + \left(5 a^{2} + 8 a + 5\right)\cdot 13^{3} + \left(7 a^{2} + 11 a + 9\right)\cdot 13^{4} + \left(4 a^{2} + a + 4\right)\cdot 13^{5} + \left(a^{2} + 4 a + 1\right)\cdot 13^{6} + \left(6 a^{2} + 6 a + 4\right)\cdot 13^{7} + \left(5 a^{2} + a + 4\right)\cdot 13^{8} + \left(7 a^{2} + 10 a + 9\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a^{2} + 8 a + 9 + \left(2 a^{2} + 4 a + 10\right)\cdot 13 + \left(3 a^{2} + 6 a + 9\right)\cdot 13^{2} + \left(11 a^{2} + 5 a + 5\right)\cdot 13^{3} + \left(10 a^{2} + 4 a + 5\right)\cdot 13^{4} + \left(11 a^{2} + 2 a + 11\right)\cdot 13^{5} + \left(11 a^{2} + 5 a + 6\right)\cdot 13^{6} + \left(3 a^{2} + 11 a + 8\right)\cdot 13^{7} + \left(4 a^{2} + 5 a + 3\right)\cdot 13^{8} + \left(5 a^{2} + 9 a + 9\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a^{2} + 6 a + 6 + 11 a^{2} 13 + \left(8 a^{2} + 3 a + 12\right)\cdot 13^{2} + \left(11 a^{2} + 4 a + 9\right)\cdot 13^{3} + \left(2 a^{2} + 11 a + 7\right)\cdot 13^{4} + \left(11 a + 7\right)\cdot 13^{5} + \left(11 a^{2} + 11 a + 5\right)\cdot 13^{6} + \left(4 a^{2} + 6 a + 2\right)\cdot 13^{7} + 2 a^{2} 13^{8} + \left(12 a^{2} + 5 a + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 a^{2} + 4 a + 6 + \left(7 a^{2} + 8 a + 12\right)\cdot 13 + \left(2 a^{2} + 7 a + 7\right)\cdot 13^{2} + \left(9 a^{2} + 6\right)\cdot 13^{3} + \left(2 a^{2} + 3 a + 7\right)\cdot 13^{4} + \left(8 a^{2} + 12 a + 9\right)\cdot 13^{5} + \left(9 a + 4\right)\cdot 13^{6} + \left(2 a^{2} + 12 a + 7\right)\cdot 13^{7} + \left(5 a^{2} + 10 a + 12\right)\cdot 13^{8} + \left(6 a^{2} + 10 a + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 3 a^{2} + 8 a + 11 + \left(8 a^{2} + 6 a + 10\right)\cdot 13 + \left(11 a^{2} + 11 a + 9\right)\cdot 13^{2} + \left(10 a^{2} + 4 a + 7\right)\cdot 13^{3} + \left(3 a^{2} + 10 a + 1\right)\cdot 13^{4} + \left(10 a + 6\right)\cdot 13^{5} + \left(2 a^{2} + 3\right)\cdot 13^{6} + \left(11 a^{2} + 4 a + 11\right)\cdot 13^{7} + \left(10 a^{2} + 10 a + 10\right)\cdot 13^{8} + \left(9 a^{2} + 8 a + 2\right)\cdot 13^{9} +O(13^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)$ | $4$ |
| $18$ | $2$ | $(1,4)(2,6)(5,8)$ | $-2$ |
| $27$ | $2$ | $(1,2)(3,7)(4,6)$ | $0$ |
| $27$ | $2$ | $(1,2)(4,6)$ | $2$ |
| $54$ | $2$ | $(1,2)(3,4)(6,7)(8,9)$ | $0$ |
| $6$ | $3$ | $(3,7,9)$ | $3$ |
| $8$ | $3$ | $(1,2,5)(3,7,9)(4,6,8)$ | $-3$ |
| $12$ | $3$ | $(3,7,9)(4,6,8)$ | $0$ |
| $72$ | $3$ | $(1,4,3)(2,6,7)(5,8,9)$ | $0$ |
| $54$ | $4$ | $(1,6,2,4)(5,8)$ | $-2$ |
| $162$ | $4$ | $(1,7,2,3)(5,9)(6,8)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,6)(3,7,9)(5,8)$ | $1$ |
| $36$ | $6$ | $(1,3,2,7,5,9)$ | $-2$ |
| $36$ | $6$ | $(1,2)(3,7,9)$ | $1$ |
| $36$ | $6$ | $(1,2)(3,7,9)(4,6,8)$ | $-2$ |
| $54$ | $6$ | $(1,2)(3,9,7)(4,6)$ | $-1$ |
| $72$ | $6$ | $(1,4,2,6,5,8)(3,7,9)$ | $1$ |
| $108$ | $6$ | $(1,2)(3,6,7,8,9,4)$ | $0$ |
| $216$ | $6$ | $(1,6,7,2,4,3)(5,8,9)$ | $0$ |
| $144$ | $9$ | $(1,4,3,2,6,7,5,8,9)$ | $0$ |
| $108$ | $12$ | $(1,6,2,4)(3,7,9)(5,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.