Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(165088694671507456\)\(\medspace = 2^{18} \cdot 229^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.3.176003742784.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | even |
| Determinant: | 1.229.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.3.176003742784.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 9x^{7} - 7x^{6} + 2x^{5} + 14x^{4} - 9x^{3} - 11x^{2} - 3x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 15 a^{2} + 4 a + 5 + \left(4 a^{2} + 7 a + 14\right)\cdot 17 + \left(6 a^{2} + 9 a + 6\right)\cdot 17^{2} + \left(6 a^{2} + 15 a + 8\right)\cdot 17^{3} + \left(12 a^{2} + 2 a + 7\right)\cdot 17^{4} + \left(7 a^{2} + 15 a + 15\right)\cdot 17^{5} + \left(7 a^{2} + 9 a + 2\right)\cdot 17^{6} + \left(6 a^{2} + 2 a + 7\right)\cdot 17^{7} + \left(3 a^{2} + 14 a + 7\right)\cdot 17^{8} + \left(9 a^{2} + 16 a\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a^{2} + 2 + \left(13 a^{2} + 2\right)\cdot 17 + \left(7 a^{2} + 10 a + 13\right)\cdot 17^{2} + \left(a^{2} + 11 a + 6\right)\cdot 17^{3} + \left(7 a + 13\right)\cdot 17^{4} + \left(16 a^{2} + 7 a + 5\right)\cdot 17^{5} + \left(6 a^{2} + 3 a + 7\right)\cdot 17^{6} + \left(10 a^{2} + 8 a\right)\cdot 17^{7} + \left(10 a^{2} + 14 a + 15\right)\cdot 17^{8} + \left(8 a^{2} + 9 a + 16\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a^{2} + 13 a + 10 + \left(8 a^{2} + 12\right)\cdot 17 + \left(15 a^{2} + 6 a + 16\right)\cdot 17^{2} + \left(13 a^{2} + 14 a + 4\right)\cdot 17^{3} + \left(2 a^{2} + 3 a + 6\right)\cdot 17^{4} + \left(12 a^{2} + 10 a + 8\right)\cdot 17^{5} + \left(12 a^{2} + 11 a + 13\right)\cdot 17^{6} + \left(14 a + 9\right)\cdot 17^{7} + \left(6 a^{2} + 13 a + 13\right)\cdot 17^{8} + \left(4 a^{2} + 6 a + 8\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 8 a + 12 + \left(5 a^{2} + 11 a + 8\right)\cdot 17 + \left(5 a^{2} + 6 a\right)\cdot 17^{2} + \left(13 a^{2} + 11 a + 13\right)\cdot 17^{3} + \left(11 a^{2} + 2 a + 12\right)\cdot 17^{4} + \left(16 a^{2} + 15 a + 15\right)\cdot 17^{5} + \left(6 a^{2} + 2 a + 13\right)\cdot 17^{6} + \left(4 a^{2} + 15 a + 5\right)\cdot 17^{7} + \left(2 a^{2} + 4 a + 12\right)\cdot 17^{8} + \left(4 a^{2} + 7 a + 2\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( a^{2} + 12 a + 8 + \left(10 a^{2} + 10 a + 2\right)\cdot 17 + \left(14 a^{2} + 10 a + 16\right)\cdot 17^{2} + \left(8 a + 1\right)\cdot 17^{3} + \left(9 a^{2} + 6 a + 16\right)\cdot 17^{4} + \left(3 a^{2} + 12 a + 13\right)\cdot 17^{5} + \left(a^{2} + 11 a + 5\right)\cdot 17^{6} + \left(14 a^{2} + 8 a + 7\right)\cdot 17^{7} + \left(6 a^{2} + 5 a + 8\right)\cdot 17^{8} + \left(8 a^{2} + 10 a + 11\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 12 a^{2} + 9 a + 4 + \left(15 a^{2} + 5 a + 6\right)\cdot 17 + \left(3 a^{2} + 3\right)\cdot 17^{2} + \left(2 a^{2} + 11 a + 14\right)\cdot 17^{3} + \left(5 a^{2} + 6 a + 7\right)\cdot 17^{4} + \left(a^{2} + 11 a + 12\right)\cdot 17^{5} + \left(3 a^{2} + 10 a + 12\right)\cdot 17^{6} + \left(2 a^{2} + 10 a + 10\right)\cdot 17^{7} + \left(4 a^{2} + 14 a + 6\right)\cdot 17^{8} + \left(4 a^{2} + 16 a + 14\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 2 a^{2} + 5 a + 2 + \left(7 a^{2} + 15 a + 10\right)\cdot 17 + 5 a^{2} 17^{2} + \left(14 a^{2} + 7 a + 8\right)\cdot 17^{3} + \left(9 a^{2} + 11 a + 11\right)\cdot 17^{4} + \left(9 a^{2} + 3 a + 16\right)\cdot 17^{5} + \left(2 a^{2} + 4 a + 10\right)\cdot 17^{6} + \left(6 a^{2} + 16 a + 12\right)\cdot 17^{7} + \left(11 a^{2} + 14 a + 12\right)\cdot 17^{8} + \left(3 a^{2} + 9 a + 13\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( a^{2} + a + 5 + \left(2 a^{2} + 16 a\right)\cdot 17 + \left(13 a^{2} + 13 a + 11\right)\cdot 17^{2} + \left(9 a^{2} + 9 a + 6\right)\cdot 17^{3} + \left(12 a^{2} + 7 a + 10\right)\cdot 17^{4} + \left(5 a^{2} + 6 a + 4\right)\cdot 17^{5} + \left(8 a^{2} + 12 a + 8\right)\cdot 17^{6} + \left(13 a^{2} + 5 a + 2\right)\cdot 17^{7} + \left(6 a^{2} + 14 a + 1\right)\cdot 17^{8} + \left(16 a^{2} + 6 a + 5\right)\cdot 17^{9} +O(17^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 11 a^{2} + 16 a + 6 + \left(a^{2} + 11\right)\cdot 17 + \left(13 a^{2} + 10 a + 16\right)\cdot 17^{2} + \left(5 a^{2} + 12 a + 3\right)\cdot 17^{3} + \left(4 a^{2} + a + 16\right)\cdot 17^{4} + \left(12 a^{2} + 3 a + 8\right)\cdot 17^{5} + \left(a^{2} + a + 9\right)\cdot 17^{6} + \left(10 a^{2} + 3 a + 11\right)\cdot 17^{7} + \left(16 a^{2} + 5 a + 7\right)\cdot 17^{8} + \left(8 a^{2} + 11\right)\cdot 17^{9} +O(17^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(3,7)$ | $4$ |
| $18$ | $2$ | $(2,3)(4,7)(6,9)$ | $2$ |
| $27$ | $2$ | $(2,4)(3,7)$ | $0$ |
| $27$ | $2$ | $(1,5)(2,4)(3,7)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,4)(5,7)(8,9)$ | $2$ |
| $6$ | $3$ | $(1,5,8)$ | $0$ |
| $8$ | $3$ | $(1,5,8)(2,4,6)(3,7,9)$ | $3$ |
| $12$ | $3$ | $(1,5,8)(3,7,9)$ | $-3$ |
| $72$ | $3$ | $(1,3,2)(4,5,7)(6,8,9)$ | $0$ |
| $54$ | $4$ | $(2,7,4,3)(6,9)$ | $0$ |
| $162$ | $4$ | $(1,3,5,7)(2,4)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,5,8)(2,3)(4,7)(6,9)$ | $2$ |
| $36$ | $6$ | $(1,7,5,9,8,3)$ | $-1$ |
| $36$ | $6$ | $(1,5,8)(3,7)$ | $-2$ |
| $36$ | $6$ | $(1,5,8)(2,4,6)(3,7)$ | $1$ |
| $54$ | $6$ | $(1,8,5)(2,4)(3,7)$ | $0$ |
| $72$ | $6$ | $(1,5,8)(2,9,6,7,4,3)$ | $-1$ |
| $108$ | $6$ | $(1,7,5,9,8,3)(2,4)$ | $-1$ |
| $216$ | $6$ | $(1,3,4,5,7,2)(6,8,9)$ | $0$ |
| $144$ | $9$ | $(1,7,4,5,9,6,8,3,2)$ | $0$ |
| $108$ | $12$ | $(1,5,8)(2,7,4,3)(6,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.