Basic invariants
| Dimension: | $4$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(241299\)\(\medspace = 3^{6} \cdot 331 \) |
| Artin stem field: | Galois closure of 9.5.79310879217.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 12T175 |
| Parity: | even |
| Determinant: | 1.993.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.5.79310879217.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} - 7x^{7} + 6x^{6} + 19x^{5} + 14x^{4} - 15x^{3} - 17x^{2} - 7x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$:
\( x^{3} + 2x + 18 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a^{2} + 2 a + 1 + \left(12 a^{2} + 13 a + 9\right)\cdot 23 + \left(15 a^{2} + 3 a + 9\right)\cdot 23^{2} + \left(14 a^{2} + 14 a + 2\right)\cdot 23^{3} + \left(4 a^{2} + 21 a + 17\right)\cdot 23^{4} + \left(5 a^{2} + 9\right)\cdot 23^{5} + \left(18 a^{2} + 22\right)\cdot 23^{6} + \left(12 a^{2} + 20 a + 14\right)\cdot 23^{7} + \left(21 a^{2} + 22 a + 3\right)\cdot 23^{8} + \left(5 a^{2} + 7 a + 12\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a^{2} + 7 a + 3 + 19 a\cdot 23 + \left(12 a^{2} + 19 a + 7\right)\cdot 23^{2} + \left(14 a^{2} + 16 a + 20\right)\cdot 23^{3} + \left(13 a^{2} + 21 a + 10\right)\cdot 23^{4} + \left(a^{2} + 9 a + 10\right)\cdot 23^{5} + \left(11 a^{2} + 6 a + 13\right)\cdot 23^{6} + \left(14 a^{2} + 13 a + 2\right)\cdot 23^{7} + \left(15 a^{2} + 10 a + 7\right)\cdot 23^{8} + \left(17 a^{2} + 18 a + 18\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a^{2} + 6 a + 19 + \left(17 a^{2} + 18 a\right)\cdot 23 + \left(15 a^{2} + 15 a + 3\right)\cdot 23^{2} + \left(2 a^{2} + 8 a + 12\right)\cdot 23^{3} + \left(4 a + 19\right)\cdot 23^{4} + \left(3 a^{2} + 2 a + 15\right)\cdot 23^{5} + \left(17 a^{2} + 19 a + 10\right)\cdot 23^{6} + \left(21 a + 4\right)\cdot 23^{7} + \left(10 a^{2} + 5 a + 6\right)\cdot 23^{8} + \left(22 a^{2} + 16 a\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a^{2} + 13 a + 20 + \left(15 a^{2} + 18 a + 12\right)\cdot 23 + \left(12 a^{2} + 18 a + 5\right)\cdot 23^{2} + \left(3 a^{2} + 4 a + 18\right)\cdot 23^{3} + \left(14 a^{2} + 11 a + 6\right)\cdot 23^{4} + \left(19 a^{2} + 14 a + 21\right)\cdot 23^{5} + \left(12 a^{2} + 9 a + 22\right)\cdot 23^{6} + \left(6 a^{2} + 6 a + 21\right)\cdot 23^{7} + \left(20 a^{2} + a + 1\right)\cdot 23^{8} + \left(3 a^{2} + 16 a + 17\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a^{2} + 21 a + 11 + \left(5 a^{2} + 17 a + 7\right)\cdot 23 + \left(3 a^{2} + 4 a + 9\right)\cdot 23^{2} + \left(16 a^{2} + 17 a + 22\right)\cdot 23^{3} + \left(2 a^{2} + 9 a + 22\right)\cdot 23^{4} + \left(3 a^{2} + 21 a + 15\right)\cdot 23^{5} + \left(17 a^{2} + 3 a + 10\right)\cdot 23^{6} + \left(15 a^{2} + 14 a + 1\right)\cdot 23^{7} + \left(10 a^{2} + 14 a + 7\right)\cdot 23^{8} + \left(19 a^{2} + 5 a + 19\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a^{2} + 19 a + 13 + \left(22 a^{2} + 9 a + 15\right)\cdot 23 + \left(3 a^{2} + 2 a + 2\right)\cdot 23^{2} + \left(4 a^{2} + 20 a + 14\right)\cdot 23^{3} + \left(20 a^{2} + 8 a + 15\right)\cdot 23^{4} + \left(16 a^{2} + 22 a + 3\right)\cdot 23^{5} + \left(11 a^{2} + 22 a + 11\right)\cdot 23^{6} + \left(6 a^{2} + 9 a + 4\right)\cdot 23^{7} + \left(2 a^{2} + 2 a + 11\right)\cdot 23^{8} + \left(4 a^{2} + a + 6\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 6 a^{2} + 8 a + 11 + \left(18 a^{2} + 14 a + 1\right)\cdot 23 + \left(17 a^{2} + 20\right)\cdot 23^{2} + \left(4 a^{2} + 4 a + 19\right)\cdot 23^{3} + \left(4 a^{2} + 13 a + 8\right)\cdot 23^{4} + \left(21 a^{2} + 7 a\right)\cdot 23^{5} + \left(14 a^{2} + 13 a + 18\right)\cdot 23^{6} + \left(3 a^{2} + 19 a + 2\right)\cdot 23^{7} + \left(4 a^{2} + 21 a + 11\right)\cdot 23^{8} + \left(13 a^{2} + 21 a + 6\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 8 a^{2} + 4 a + 17 + \left(22 a + 7\right)\cdot 23 + \left(2 a^{2} + 16 a + 1\right)\cdot 23^{2} + \left(a^{2} + 13 a + 10\right)\cdot 23^{3} + \left(8 a^{2} + 3 a + 3\right)\cdot 23^{4} + \left(22 a^{2} + 13 a + 15\right)\cdot 23^{5} + \left(16 a^{2} + 17 a + 13\right)\cdot 23^{6} + \left(8 a^{2} + 9 a + 2\right)\cdot 23^{7} + \left(12 a^{2} + 11 a + 18\right)\cdot 23^{8} + \left(14 a^{2} + 16 a + 21\right)\cdot 23^{9} +O(23^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 6 a^{2} + 12 a + 22 + \left(22 a^{2} + 4 a + 13\right)\cdot 23 + \left(8 a^{2} + 9 a + 10\right)\cdot 23^{2} + \left(7 a^{2} + 15 a + 18\right)\cdot 23^{3} + \left(a^{2} + 20 a + 9\right)\cdot 23^{4} + \left(22 a^{2} + 22 a + 22\right)\cdot 23^{5} + \left(17 a^{2} + 21 a + 14\right)\cdot 23^{6} + \left(22 a^{2} + 22 a + 13\right)\cdot 23^{7} + \left(17 a^{2} + 2\right)\cdot 23^{8} + \left(13 a^{2} + 11 a + 13\right)\cdot 23^{9} +O(23^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $18$ | $2$ | $(1,5)(3,4)(8,9)$ | $2$ |
| $27$ | $2$ | $(2,6)(3,9)$ | $0$ |
| $4$ | $3$ | $(1,9,3)(2,7,6)(4,5,8)$ | $3 \zeta_{3} + 1$ |
| $4$ | $3$ | $(1,3,9)(2,6,7)(4,8,5)$ | $-3 \zeta_{3} - 2$ |
| $6$ | $3$ | $(4,5,8)$ | $-2$ |
| $12$ | $3$ | $(2,7,6)(4,5,8)$ | $1$ |
| $72$ | $3$ | $(1,5,2)(3,8,7)(4,6,9)$ | $1$ |
| $162$ | $4$ | $(1,4)(3,5,9,8)(6,7)$ | $0$ |
| $18$ | $6$ | $(1,6)(2,9)(3,7)(4,8,5)$ | $2 \zeta_{3}$ |
| $18$ | $6$ | $(1,6)(2,9)(3,7)(4,5,8)$ | $-2 \zeta_{3} - 2$ |
| $36$ | $6$ | $(1,4,9,5,3,8)(2,7,6)$ | $\zeta_{3} + 1$ |
| $36$ | $6$ | $(1,8,3,5,9,4)(2,6,7)$ | $-\zeta_{3}$ |
| $36$ | $6$ | $(2,8,7,4,6,5)$ | $-1$ |
| $54$ | $6$ | $(2,6)(3,9)(4,8,5)$ | $0$ |
| $72$ | $9$ | $(1,6,8,9,2,4,3,7,5)$ | $-\zeta_{3} - 1$ |
| $72$ | $9$ | $(1,8,2,3,5,6,9,4,7)$ | $\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.