Basic invariants
| Dimension: | $8$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(1185607388736\)\(\medspace = 2^{6} \cdot 3^{6} \cdot 71^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.664071871846464.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T178 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.1.664071871846464.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} - 2x^{7} + 23x^{6} - 72x^{5} + 243x^{4} - 529x^{3} + 510x^{2} - 172x + 28 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{3} + x + 28 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 20 a + 25 + \left(9 a^{2} + 20 a + 28\right)\cdot 31 + \left(28 a^{2} + 19 a + 18\right)\cdot 31^{2} + \left(8 a^{2} + 26\right)\cdot 31^{3} + \left(19 a^{2} + 28 a + 8\right)\cdot 31^{4} + \left(12 a^{2} + 25 a + 16\right)\cdot 31^{5} + \left(25 a^{2} + 24 a + 2\right)\cdot 31^{6} + \left(29 a^{2} + 18 a + 5\right)\cdot 31^{7} + \left(27 a^{2} + 25 a + 7\right)\cdot 31^{8} + \left(30 a^{2} + a + 30\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 a^{2} + 12 a + 8 + \left(a^{2} + 28 a + 3\right)\cdot 31 + \left(25 a^{2} + 26 a + 26\right)\cdot 31^{2} + \left(23 a + 8\right)\cdot 31^{3} + \left(12 a^{2} + 5\right)\cdot 31^{4} + \left(4 a^{2} + 6 a + 14\right)\cdot 31^{5} + \left(27 a^{2} + 16 a + 16\right)\cdot 31^{6} + \left(6 a^{2} + 6 a + 3\right)\cdot 31^{7} + \left(20 a^{2} + 2 a + 28\right)\cdot 31^{8} + \left(21 a^{2} + 9 a + 24\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 a^{2} + 13 a + 24 + \left(23 a^{2} + 21 a + 28\right)\cdot 31 + \left(26 a^{2} + 28 a + 16\right)\cdot 31^{2} + \left(13 a^{2} + 4 a + 17\right)\cdot 31^{3} + \left(4 a^{2} + 12 a + 10\right)\cdot 31^{4} + \left(18 a^{2} + 9 a + 23\right)\cdot 31^{5} + \left(8 a^{2} + 28 a + 24\right)\cdot 31^{6} + \left(9 a^{2} + 25 a + 25\right)\cdot 31^{7} + \left(17 a^{2} + 19 a + 15\right)\cdot 31^{8} + \left(26 a^{2} + 25 a + 7\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 a^{2} + 15 a + 17 + \left(27 a^{2} + 16 a + 3\right)\cdot 31 + \left(28 a^{2} + 21 a + 20\right)\cdot 31^{2} + \left(3 a^{2} + 12 a + 4\right)\cdot 31^{3} + \left(6 a^{2} + 29 a + 21\right)\cdot 31^{4} + \left(17 a^{2} + a + 2\right)\cdot 31^{5} + \left(22 a^{2} + 30 a\right)\cdot 31^{6} + \left(16 a^{2} + 24 a + 27\right)\cdot 31^{7} + \left(22 a^{2} + 20 a + 11\right)\cdot 31^{8} + \left(13 a^{2} + 2 a + 30\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a^{2} + 23 a + \left(28 a^{2} + 26 a + 4\right)\cdot 31 + \left(23 a^{2} + 19 a + 27\right)\cdot 31^{2} + \left(15 a^{2} + 13 a + 22\right)\cdot 31^{3} + \left(5 a^{2} + 15 a + 20\right)\cdot 31^{4} + \left(14 a^{2} + 7 a\right)\cdot 31^{5} + \left(28 a^{2} + 16 a + 4\right)\cdot 31^{6} + \left(26 a^{2} + 22 a + 13\right)\cdot 31^{7} + \left(22 a^{2} + 23 a + 22\right)\cdot 31^{8} + \left(9 a + 21\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 10 a^{2} + 20 a + 16 + \left(6 a^{2} + 16 a + 16\right)\cdot 31 + \left(5 a^{2} + 3 a + 3\right)\cdot 31^{2} + \left(18 a^{2} + 3 a + 12\right)\cdot 31^{3} + \left(20 a^{2} + a + 30\right)\cdot 31^{4} + \left(22 a^{2} + 12 a + 22\right)\cdot 31^{5} + \left(30 a^{2} + 3 a + 26\right)\cdot 31^{6} + \left(18 a^{2} + 11 a + 28\right)\cdot 31^{7} + \left(18 a^{2} + 6 a\right)\cdot 31^{8} + \left(21 a^{2} + 11 a + 24\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 13 a^{2} + 24 a + 6 + \left(6 a^{2} + 18 a + 10\right)\cdot 31 + \left(9 a^{2} + 20 a + 17\right)\cdot 31^{2} + \left(11 a^{2} + 4 a + 9\right)\cdot 31^{3} + \left(19 a^{2} + 17 a + 9\right)\cdot 31^{4} + \left(30 a^{2} + 21 a + 1\right)\cdot 31^{5} + \left(10 a^{2} + 15 a + 13\right)\cdot 31^{6} + \left(18 a^{2} + 14 a + 7\right)\cdot 31^{7} + \left(16 a^{2} + 17 a + 18\right)\cdot 31^{8} + \left(16 a^{2} + 18 a + 11\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 13 a^{2} + 22 a + 18 + \left(15 a^{2} + 24 a + 22\right)\cdot 31 + \left(28 a^{2} + 7 a + 8\right)\cdot 31^{2} + \left(3 a^{2} + 27 a + 23\right)\cdot 31^{3} + \left(22 a^{2} + a + 10\right)\cdot 31^{4} + \left(26 a^{2} + 24 a + 15\right)\cdot 31^{5} + \left(5 a^{2} + 2 a + 20\right)\cdot 31^{6} + \left(13 a^{2} + a + 14\right)\cdot 31^{7} + \left(15 a^{2} + 30 a + 19\right)\cdot 31^{8} + \left(9 a^{2} + 17 a + 5\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 28 a^{2} + 6 a + 13 + \left(5 a^{2} + 12 a + 6\right)\cdot 31 + \left(10 a^{2} + 6 a + 16\right)\cdot 31^{2} + \left(16 a^{2} + 2 a + 29\right)\cdot 31^{3} + \left(14 a^{2} + 18 a + 6\right)\cdot 31^{4} + \left(8 a^{2} + 15 a + 27\right)\cdot 31^{5} + \left(26 a^{2} + 17 a + 15\right)\cdot 31^{6} + \left(14 a^{2} + 29 a + 29\right)\cdot 31^{7} + \left(24 a^{2} + 8 a + 30\right)\cdot 31^{8} + \left(13 a^{2} + 27 a + 29\right)\cdot 31^{9} +O(31^{10})\)
|
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 value |
| $1$ | $1$ | $()$ | $8$ |
| $27$ | $2$ | $(1,2)(3,7)$ | $0$ |
| $54$ | $2$ | $(1,2)(3,5)(6,7)(8,9)$ | $0$ |
| $6$ | $3$ | $(1,4,2)$ | $-4$ |
| $8$ | $3$ | $(1,2,4)(3,7,8)(5,6,9)$ | $-1$ |
| $12$ | $3$ | $(1,4,2)(3,8,7)$ | $2$ |
| $72$ | $3$ | $(1,5,3)(2,6,7)(4,9,8)$ | $2$ |
| $54$ | $4$ | $(1,7,2,3)(4,8)$ | $0$ |
| $54$ | $6$ | $(1,2,4)(3,7)(5,6)$ | $0$ |
| $108$ | $6$ | $(1,8,4,7,2,3)(5,6)$ | $0$ |
| $72$ | $9$ | $(1,6,7,2,9,8,4,5,3)$ | $-1$ |
| $72$ | $9$ | $(1,9,8,4,6,7,2,5,3)$ | $-1$ |
| $54$ | $12$ | $(1,4,2)(3,5,7,6)(8,9)$ | $0$ |
| $54$ | $12$ | $(1,2,4)(3,5,7,6)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.