Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(11670913024\)\(\medspace = 2^{18} \cdot 211^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.2404846336.1 |
| 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 9.1.2404846336.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 4x^{7} - 2x^{5} + 8x^{4} - x^{3} + 2x^{2} + 4x + 2 \)
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The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$:
\( x^{3} + 3x + 51 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 32 a^{2} + 47 a + 27 + \left(47 a^{2} + a + 6\right)\cdot 53 + \left(41 a^{2} + 17 a + 7\right)\cdot 53^{2} + \left(46 a^{2} + 37 a + 4\right)\cdot 53^{3} + \left(47 a + 14\right)\cdot 53^{4} + \left(11 a^{2} + 7 a + 26\right)\cdot 53^{5} + \left(2 a^{2} + 23 a + 52\right)\cdot 53^{6} + \left(39 a^{2} + 10 a + 7\right)\cdot 53^{7} + \left(16 a^{2} + 18 a + 42\right)\cdot 53^{8} + \left(49 a^{2} + 16 a + 15\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 a^{2} + 13 a + 47 + \left(6 a^{2} + 42 a + 12\right)\cdot 53 + \left(29 a^{2} + 31 a + 42\right)\cdot 53^{2} + \left(19 a^{2} + 14 a + 41\right)\cdot 53^{3} + \left(51 a^{2} + 2 a + 21\right)\cdot 53^{4} + \left(19 a^{2} + 47 a + 33\right)\cdot 53^{5} + \left(46 a^{2} + 17 a + 10\right)\cdot 53^{6} + \left(43 a^{2} + a + 25\right)\cdot 53^{7} + \left(23 a^{2} + 45 a + 5\right)\cdot 53^{8} + \left(12 a^{2} + 3 a + 42\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 + 10\cdot 53 + 22\cdot 53^{2} + 2\cdot 53^{3} + 21\cdot 53^{4} + 10\cdot 53^{5} + 13\cdot 53^{6} + 40\cdot 53^{7} + 7\cdot 53^{8} + 45\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a^{2} + 35 a + 50 + \left(26 a^{2} + 11 a + 51\right)\cdot 53 + \left(12 a^{2} + 28 a + 8\right)\cdot 53^{2} + \left(21 a^{2} + 33 a + 45\right)\cdot 53^{3} + \left(8 a^{2} + 48 a + 41\right)\cdot 53^{4} + \left(22 a^{2} + 6 a + 37\right)\cdot 53^{5} + \left(16 a^{2} + 37 a + 3\right)\cdot 53^{6} + \left(33 a^{2} + 4 a + 4\right)\cdot 53^{7} + \left(17 a^{2} + 38 a + 46\right)\cdot 53^{8} + \left(5 a^{2} + 34 a + 27\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 48 a^{2} + 15 a + 6 + \left(16 a^{2} + 29 a + 51\right)\cdot 53 + \left(43 a^{2} + 44 a + 9\right)\cdot 53^{2} + \left(50 a^{2} + 35 a + 12\right)\cdot 53^{3} + \left(a^{2} + 15 a + 16\right)\cdot 53^{4} + \left(11 a^{2} + 16 a + 26\right)\cdot 53^{5} + \left(18 a^{2} + 45 a + 31\right)\cdot 53^{6} + \left(50 a^{2} + 8 a + 30\right)\cdot 53^{7} + \left(17 a^{2} + 28 a + 44\right)\cdot 53^{8} + \left(38 a^{2} + 7 a + 46\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 a^{2} + 44 a + 15 + \left(41 a^{2} + 21 a + 47\right)\cdot 53 + \left(20 a^{2} + 44 a + 17\right)\cdot 53^{2} + \left(8 a^{2} + 32 a + 33\right)\cdot 53^{3} + \left(50 a^{2} + 42 a + 6\right)\cdot 53^{4} + \left(30 a^{2} + 28 a + 13\right)\cdot 53^{5} + \left(32 a^{2} + 37 a + 7\right)\cdot 53^{6} + \left(16 a^{2} + 33 a + 16\right)\cdot 53^{7} + \left(18 a^{2} + 6 a + 45\right)\cdot 53^{8} + \left(18 a^{2} + 29 a + 6\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 38 + 44\cdot 53 + 9\cdot 53^{2} + 19\cdot 53^{3} + 29\cdot 53^{4} + 26\cdot 53^{5} + 53^{6} + 24\cdot 53^{7} + 26\cdot 53^{8} + 49\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 10 a^{2} + 5 a + 52 + \left(20 a^{2} + 52 a + 39\right)\cdot 53 + \left(11 a^{2} + 45 a + 6\right)\cdot 53^{2} + \left(12 a^{2} + 4 a + 27\right)\cdot 53^{3} + \left(46 a^{2} + 2 a + 11\right)\cdot 53^{4} + \left(10 a^{2} + 52 a + 15\right)\cdot 53^{5} + \left(43 a^{2} + 50 a + 4\right)\cdot 53^{6} + \left(28 a^{2} + 46 a + 48\right)\cdot 53^{7} + \left(11 a^{2} + 22 a + 33\right)\cdot 53^{8} + \left(35 a^{2} + 14 a + 34\right)\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 29 + 34\cdot 53^{2} + 26\cdot 53^{3} + 49\cdot 53^{4} + 22\cdot 53^{5} + 34\cdot 53^{6} + 15\cdot 53^{7} + 13\cdot 53^{8} + 49\cdot 53^{9} +O(53^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,5)$ | $0$ |
| $18$ | $2$ | $(1,2)(4,5)(6,8)$ | $4$ |
| $27$ | $2$ | $(1,5)(2,4)(3,7)$ | $0$ |
| $27$ | $2$ | $(1,5)(3,7)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,4)(5,7)(6,9)$ | $0$ |
| $6$ | $3$ | $(3,7,9)$ | $-4$ |
| $8$ | $3$ | $(1,6,5)(2,8,4)(3,9,7)$ | $-1$ |
| $12$ | $3$ | $(1,6,5)(3,9,7)$ | $2$ |
| $72$ | $3$ | $(1,2,3)(4,7,5)(6,8,9)$ | $2$ |
| $54$ | $4$ | $(1,3,5,7)(6,9)$ | $0$ |
| $162$ | $4$ | $(1,3,5,7)(2,4)(6,9)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,7,9)(4,5)(6,8)$ | $-2$ |
| $36$ | $6$ | $(1,3,6,9,5,7)$ | $-2$ |
| $36$ | $6$ | $(1,5)(3,7,9)$ | $0$ |
| $36$ | $6$ | $(1,5)(2,4,8)(3,7,9)$ | $0$ |
| $54$ | $6$ | $(1,5)(2,4)(3,9,7)$ | $0$ |
| $72$ | $6$ | $(1,2,6,8,5,4)(3,7,9)$ | $1$ |
| $108$ | $6$ | $(1,3,6,9,5,7)(2,4)$ | $0$ |
| $216$ | $6$ | $(1,2,3,5,4,7)(6,8,9)$ | $0$ |
| $144$ | $9$ | $(1,2,3,6,8,9,5,4,7)$ | $-1$ |
| $108$ | $12$ | $(1,2,5,4)(3,7,9)(6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.