Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(86920128\)\(\medspace = 2^{6} \cdot 3^{10} \cdot 23 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.9387373824.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T319 |
| Parity: | odd |
| Determinant: | 1.23.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.9387373824.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 6x^{7} - 6x^{6} + 6x^{5} + 3x + 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 11 a^{2} + a + 11 + \left(4 a^{2} + a + 12\right)\cdot 13 + \left(5 a^{2} + 4 a + 7\right)\cdot 13^{2} + \left(4 a^{2} + 2 a + 7\right)\cdot 13^{3} + \left(6 a^{2} + 4 a + 12\right)\cdot 13^{4} + \left(7 a + 8\right)\cdot 13^{5} + \left(6 a^{2} + 12\right)\cdot 13^{6} + \left(12 a^{2} + 4 a + 4\right)\cdot 13^{7} + \left(9 a^{2} + a + 1\right)\cdot 13^{8} + \left(10 a^{2} + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a^{2} + 8 a + 12 + \left(12 a^{2} + 2 a\right)\cdot 13 + \left(4 a^{2} + a + 5\right)\cdot 13^{2} + \left(12 a^{2} + 11 a + 7\right)\cdot 13^{3} + \left(4 a^{2} + 8 a + 3\right)\cdot 13^{4} + \left(7 a^{2} + 11 a + 2\right)\cdot 13^{5} + \left(9 a^{2} + 9 a + 6\right)\cdot 13^{6} + \left(6 a^{2} + 4 a + 3\right)\cdot 13^{7} + \left(5 a^{2} + a + 6\right)\cdot 13^{8} + \left(3 a^{2} + 2 a + 5\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a^{2} + 8 a + 1 + \left(9 a^{2} + 3 a + 5\right)\cdot 13 + \left(6 a^{2} + 11 a + 5\right)\cdot 13^{2} + \left(2 a^{2} + 6 a + 6\right)\cdot 13^{3} + \left(8 a^{2} + 12 a + 5\right)\cdot 13^{4} + \left(5 a^{2} + 8 a + 2\right)\cdot 13^{5} + \left(8 a^{2} + 12 a\right)\cdot 13^{6} + \left(5 a^{2} + 2 a + 3\right)\cdot 13^{7} + \left(10 a^{2} + 9 a + 5\right)\cdot 13^{8} + \left(7 a^{2} + 11 a + 12\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a^{2} + 3 a + 10 + \left(3 a + 2\right)\cdot 13 + \left(11 a^{2} + 2 a + 11\right)\cdot 13^{2} + \left(6 a^{2} + 3 a + 10\right)\cdot 13^{3} + \left(7 a^{2} + 11 a + 9\right)\cdot 13^{4} + \left(2 a^{2} + 11 a + 11\right)\cdot 13^{5} + \left(3 a^{2} + 3 a + 8\right)\cdot 13^{6} + \left(2 a^{2} + 8\right)\cdot 13^{7} + \left(a^{2} + 2\right)\cdot 13^{8} + \left(8 a + 10\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a^{2} + 8 a + 6 + \left(9 a^{2} + 4 a + 1\right)\cdot 13 + \left(6 a^{2} + 3 a + 3\right)\cdot 13^{2} + \left(9 a^{2} + a + 12\right)\cdot 13^{3} + \left(10 a^{2} + 4 a + 6\right)\cdot 13^{4} + \left(a^{2} + a + 3\right)\cdot 13^{5} + \left(2 a^{2} + 8 a + 9\right)\cdot 13^{6} + \left(7 a^{2} + 4 a + 12\right)\cdot 13^{7} + \left(6 a^{2} + 12 a + 11\right)\cdot 13^{8} + \left(a^{2} + 4 a + 2\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 10 a + 11 + \left(8 a^{2} + 3\right)\cdot 13 + \left(4 a^{2} + 9 a + 11\right)\cdot 13^{2} + \left(11 a^{2} + 7 a\right)\cdot 13^{3} + \left(10 a^{2} + 8 a + 9\right)\cdot 13^{4} + \left(6 a^{2} + 8 a + 12\right)\cdot 13^{5} + \left(12 a^{2} + 9\right)\cdot 13^{6} + \left(9 a^{2} + 12 a + 8\right)\cdot 13^{7} + \left(9 a^{2} + 2 a + 8\right)\cdot 13^{8} + \left(5 a^{2} + 5 a + 9\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 a^{2} + 8 a + 7 + \left(7 a^{2} + 8 a + 6\right)\cdot 13 + \left(a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(12 a^{2} + 11 a + 1\right)\cdot 13^{3} + \left(6 a^{2} + 4 a + 8\right)\cdot 13^{4} + \left(8 a + 8\right)\cdot 13^{5} + \left(5 a^{2} + 12 a + 8\right)\cdot 13^{6} + \left(10 a^{2} + 10 a\right)\cdot 13^{7} + \left(5 a^{2} + 12\right)\cdot 13^{8} + \left(12 a^{2} + 9 a + 9\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 8 a^{2} + 9 a + 7 + \left(7 a^{2} + 8 a + 3\right)\cdot 13 + \left(9 a^{2} + 6 a + 9\right)\cdot 13^{2} + \left(a^{2} + 7 a + 12\right)\cdot 13^{3} + \left(12 a^{2} + 10 a + 2\right)\cdot 13^{4} + \left(9 a^{2} + 6 a + 4\right)\cdot 13^{5} + \left(3 a^{2} + 8 a + 5\right)\cdot 13^{6} + \left(11 a^{2} + 8 a + 3\right)\cdot 13^{7} + \left(a^{2} + 11 a + 12\right)\cdot 13^{8} + \left(2 a^{2} + 4 a + 12\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 10 a^{2} + 10 a + 3 + \left(3 a^{2} + 5 a + 2\right)\cdot 13 + \left(a^{2} + 8 a\right)\cdot 13^{2} + \left(4 a^{2} + 5\right)\cdot 13^{3} + \left(10 a^{2} + 6\right)\cdot 13^{4} + \left(3 a^{2} + 10\right)\cdot 13^{5} + \left(a^{2} + 8 a + 3\right)\cdot 13^{6} + \left(12 a^{2} + 3 a + 6\right)\cdot 13^{7} + \left(12 a + 4\right)\cdot 13^{8} + \left(8 a^{2} + 5 a + 7\right)\cdot 13^{9} +O(13^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,7)$ | $4$ |
| $18$ | $2$ | $(1,2)(3,7)(5,8)$ | $-2$ |
| $27$ | $2$ | $(1,3)(2,7)(4,6)$ | $0$ |
| $27$ | $2$ | $(1,3)(2,7)$ | $2$ |
| $54$ | $2$ | $(1,4)(2,7)(3,6)(5,9)$ | $0$ |
| $6$ | $3$ | $(4,6,9)$ | $3$ |
| $8$ | $3$ | $(1,3,5)(2,7,8)(4,6,9)$ | $-3$ |
| $12$ | $3$ | $(1,3,5)(4,6,9)$ | $0$ |
| $72$ | $3$ | $(1,4,2)(3,6,7)(5,9,8)$ | $0$ |
| $54$ | $4$ | $(1,2,3,7)(5,8)$ | $-2$ |
| $162$ | $4$ | $(2,6,7,4)(3,5)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,7)(4,6,9)(5,8)$ | $1$ |
| $36$ | $6$ | $(2,4,7,6,8,9)$ | $-2$ |
| $36$ | $6$ | $(2,7)(4,6,9)$ | $1$ |
| $36$ | $6$ | $(1,3,5)(2,7)(4,6,9)$ | $-2$ |
| $54$ | $6$ | $(1,3)(2,7)(4,9,6)$ | $-1$ |
| $72$ | $6$ | $(1,7,3,8,5,2)(4,6,9)$ | $1$ |
| $108$ | $6$ | $(1,4,3,6,5,9)(2,7)$ | $0$ |
| $216$ | $6$ | $(1,4,2,3,6,7)(5,9,8)$ | $0$ |
| $144$ | $9$ | $(1,4,7,3,6,8,5,9,2)$ | $0$ |
| $108$ | $12$ | $(1,2,3,7)(4,6,9)(5,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.