Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(32529729600\)\(\medspace = 2^{6} \cdot 3^{6} \cdot 5^{2} \cdot 167^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.3.3018036024000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.3.3018036024000.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{3} + x + 28 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 26 a^{2} + 3 a + 28 + \left(2 a^{2} + 8 a + 21\right)\cdot 31 + \left(23 a^{2} + 23 a + 22\right)\cdot 31^{2} + \left(12 a^{2} + 26 a + 25\right)\cdot 31^{3} + \left(6 a^{2} + 18 a + 4\right)\cdot 31^{4} + \left(25 a^{2} + 19 a + 14\right)\cdot 31^{5} + \left(14 a^{2} + 8 a + 10\right)\cdot 31^{6} + \left(5 a^{2} + 17 a\right)\cdot 31^{7} + \left(15 a + 21\right)\cdot 31^{8} + \left(2 a^{2} + 13 a + 14\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a^{2} + 2 a + 24 + \left(14 a^{2} + 14 a + 15\right)\cdot 31 + \left(20 a^{2} + a + 8\right)\cdot 31^{2} + \left(a^{2} + 5 a + 29\right)\cdot 31^{3} + \left(9 a^{2} + 28 a + 25\right)\cdot 31^{4} + \left(17 a^{2} + 22 a + 17\right)\cdot 31^{5} + \left(20 a^{2} + 21 a + 2\right)\cdot 31^{6} + \left(12 a^{2} + 3 a + 26\right)\cdot 31^{7} + \left(21 a^{2} + 3 a + 19\right)\cdot 31^{8} + \left(24 a^{2} + 28 a + 9\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a^{2} + 13 a + 7 + \left(a^{2} + 23 a + 26\right)\cdot 31 + \left(18 a^{2} + 11 a + 9\right)\cdot 31^{2} + \left(6 a^{2} + 5 a\right)\cdot 31^{3} + \left(11 a^{2} + 27 a + 18\right)\cdot 31^{4} + \left(6 a^{2} + 9 a\right)\cdot 31^{5} + \left(21 a^{2} + 7 a + 4\right)\cdot 31^{6} + \left(13 a^{2} + 19 a + 5\right)\cdot 31^{7} + \left(20 a^{2} + 17 a + 18\right)\cdot 31^{8} + \left(13 a^{2} + 20 a + 2\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 a^{2} + 23 a + 5 + \left(28 a^{2} + 21 a + 13\right)\cdot 31 + \left(6 a^{2} + 28 a + 2\right)\cdot 31^{2} + \left(25 a^{2} + 25 a + 23\right)\cdot 31^{3} + \left(3 a^{2} + 10 a + 2\right)\cdot 31^{4} + \left(7 a^{2} + 23 a + 1\right)\cdot 31^{5} + \left(9 a^{2} + 12 a + 27\right)\cdot 31^{6} + \left(3 a^{2} + 6 a + 18\right)\cdot 31^{7} + \left(19 a^{2} + 8 a + 27\right)\cdot 31^{8} + \left(29 a^{2} + 28 a + 2\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 10 a^{2} + 28 a + 6 + \left(13 a^{2} + 13 a + 15\right)\cdot 31 + \left(15 a^{2} + 18 a + 15\right)\cdot 31^{2} + \left(15 a^{2} + 10 a + 7\right)\cdot 31^{3} + \left(12 a^{2} + 29 a + 28\right)\cdot 31^{4} + \left(21 a^{2} + a + 30\right)\cdot 31^{5} + \left(24 a^{2} + 6 a + 25\right)\cdot 31^{6} + \left(28 a^{2} + 26 a + 5\right)\cdot 31^{7} + \left(16 a^{2} + 19 a + 27\right)\cdot 31^{8} + \left(20 a^{2} + 19 a + 6\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a^{2} + 3 a + 22 + \left(23 a^{2} + 27 a + 4\right)\cdot 31 + \left(4 a^{2} + 7 a\right)\cdot 31^{2} + \left(5 a^{2} + 12 a\right)\cdot 31^{3} + \left(11 a^{2} + 4 a + 8\right)\cdot 31^{4} + \left(7 a^{2} + 15 a + 2\right)\cdot 31^{5} + \left(5 a^{2} + 13 a + 4\right)\cdot 31^{6} + \left(28 a^{2} + 22 a + 5\right)\cdot 31^{7} + \left(11 a^{2} + 13 a + 8\right)\cdot 31^{8} + \left(8 a^{2} + 19 a + 29\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 19 a^{2} + 25 a + 13 + \left(4 a^{2} + 26 a + 2\right)\cdot 31 + \left(3 a^{2} + 30 a + 30\right)\cdot 31^{2} + \left(13 a^{2} + 22 a + 25\right)\cdot 31^{3} + \left(13 a^{2} + 7 a + 19\right)\cdot 31^{4} + \left(29 a^{2} + 27 a + 6\right)\cdot 31^{5} + \left(10 a^{2} + 8 a + 18\right)\cdot 31^{6} + \left(28 a^{2} + 22 a + 15\right)\cdot 31^{7} + \left(18 a^{2} + a + 2\right)\cdot 31^{8} + \left(20 a^{2} + 29 a + 27\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 15 a^{2} + a + 30 + \left(3 a^{2} + 3 a + 18\right)\cdot 31 + \left(26 a^{2} + 11 a + 22\right)\cdot 31^{2} + \left(13 a^{2} + 15 a + 16\right)\cdot 31^{3} + \left(9 a^{2} + 4 a + 5\right)\cdot 31^{4} + \left(23 a^{2} + 6 a + 1\right)\cdot 31^{5} + \left(16 a^{2} + 3 a\right)\cdot 31^{6} + \left(20 a^{2} + a + 21\right)\cdot 31^{7} + \left(23 a^{2} + 8 a\right)\cdot 31^{8} + \left(16 a^{2} + 14 a + 25\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 16 a^{2} + 26 a + 22 + \left(a^{2} + 16 a + 5\right)\cdot 31 + \left(6 a^{2} + 21 a + 12\right)\cdot 31^{2} + \left(30 a^{2} + 30 a + 26\right)\cdot 31^{3} + \left(15 a^{2} + 23 a + 10\right)\cdot 31^{4} + \left(17 a^{2} + 28 a + 18\right)\cdot 31^{5} + 10 a\cdot 31^{6} + \left(14 a^{2} + 5 a + 26\right)\cdot 31^{7} + \left(22 a^{2} + 5 a + 29\right)\cdot 31^{8} + \left(18 a^{2} + 13 a + 5\right)\cdot 31^{9} +O(31^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(1,4)$ | $0$ |
| $18$ | $2$ | $(1,2)(3,4)(7,8)$ | $4$ |
| $27$ | $2$ | $(1,4)(2,3)(5,6)$ | $0$ |
| $27$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $54$ | $2$ | $(1,4)(2,5)(3,6)(7,9)$ | $0$ |
| $6$ | $3$ | $(5,6,9)$ | $-4$ |
| $8$ | $3$ | $(1,4,8)(2,3,7)(5,6,9)$ | $-1$ |
| $12$ | $3$ | $(1,4,8)(5,6,9)$ | $2$ |
| $72$ | $3$ | $(1,2,5)(3,6,4)(7,9,8)$ | $2$ |
| $54$ | $4$ | $(1,3,4,2)(7,8)$ | $0$ |
| $162$ | $4$ | $(1,6,4,5)(3,7)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,4)(5,6,9)(7,8)$ | $-2$ |
| $36$ | $6$ | $(1,5,4,6,8,9)$ | $-2$ |
| $36$ | $6$ | $(1,4)(5,6,9)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,3,7)(5,6,9)$ | $0$ |
| $54$ | $6$ | $(1,4)(2,3)(5,9,6)$ | $0$ |
| $72$ | $6$ | $(1,2,4,3,8,7)(5,6,9)$ | $1$ |
| $108$ | $6$ | $(1,4)(2,5,3,6,7,9)$ | $0$ |
| $216$ | $6$ | $(1,3,6,4,2,5)(7,9,8)$ | $0$ |
| $144$ | $9$ | $(1,2,5,4,3,6,8,7,9)$ | $-1$ |
| $108$ | $12$ | $(1,3,4,2)(5,6,9)(7,8)$ | $0$ |