Basic invariants
| Dimension: | $4$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(129472\)\(\medspace = 2^{6} \cdot 7 \cdot 17^{2} \) |
| Artin stem field: | Galois closure of 9.5.8477886635008.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 12T175 |
| Parity: | even |
| Determinant: | 1.28.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.5.8477886635008.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{7} - 6x^{6} - 27x^{5} + 8x^{4} - 16x^{3} + 54x^{2} - 8x - 8 \)
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The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$:
\( x^{3} + 5x + 57 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 a^{2} + 21 a + 16 + \left(56 a^{2} + 58 a + 52\right)\cdot 59 + \left(17 a^{2} + 41 a + 20\right)\cdot 59^{2} + \left(9 a^{2} + 44 a + 43\right)\cdot 59^{3} + \left(3 a^{2} + 45 a + 36\right)\cdot 59^{4} + \left(24 a^{2} + 40 a + 15\right)\cdot 59^{5} + \left(50 a^{2} + 12 a + 45\right)\cdot 59^{6} + \left(17 a^{2} + 20 a + 55\right)\cdot 59^{7} + \left(14 a^{2} + 4 a\right)\cdot 59^{8} + \left(13 a^{2} + 51 a + 46\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 20 a^{2} + 53 a + 14 + \left(28 a^{2} + 49 a + 3\right)\cdot 59 + \left(49 a^{2} + 35 a + 7\right)\cdot 59^{2} + \left(58 a^{2} + 35 a + 15\right)\cdot 59^{3} + \left(38 a^{2} + 41 a + 1\right)\cdot 59^{4} + \left(3 a^{2} + 19 a + 44\right)\cdot 59^{5} + \left(37 a^{2} + 17 a + 58\right)\cdot 59^{6} + \left(53 a^{2} + 14 a + 46\right)\cdot 59^{7} + \left(41 a^{2} + 18 a + 12\right)\cdot 59^{8} + \left(21 a^{2} + 21 a + 45\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a^{2} + 37 a + 46 + \left(30 a^{2} + 45 a + 44\right)\cdot 59 + \left(38 a^{2} + 51 a + 10\right)\cdot 59^{2} + \left(8 a^{2} + 41 a + 41\right)\cdot 59^{3} + \left(9 a^{2} + 44 a + 56\right)\cdot 59^{4} + \left(30 a^{2} + 17 a + 35\right)\cdot 59^{5} + \left(33 a^{2} + 40 a + 8\right)\cdot 59^{6} + \left(41 a^{2} + 53 a + 56\right)\cdot 59^{7} + \left(43 a^{2} + 44 a + 39\right)\cdot 59^{8} + \left(57 a^{2} + 7 a + 56\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 46 a^{2} + a + 25 + \left(31 a^{2} + 14 a + 30\right)\cdot 59 + \left(2 a^{2} + 24 a + 28\right)\cdot 59^{2} + \left(41 a^{2} + 31 a + 11\right)\cdot 59^{3} + \left(46 a^{2} + 27 a + 44\right)\cdot 59^{4} + \left(4 a^{2} + 49\right)\cdot 59^{5} + \left(34 a^{2} + 6 a + 49\right)\cdot 59^{6} + \left(58 a^{2} + 44 a + 53\right)\cdot 59^{7} + \left(9 a + 54\right)\cdot 59^{8} + \left(47 a^{2} + 20\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 a^{2} + 44 a + 29 + \left(33 a^{2} + 9 a + 3\right)\cdot 59 + \left(50 a^{2} + 40 a + 31\right)\cdot 59^{2} + \left(49 a^{2} + 37 a\right)\cdot 59^{3} + \left(16 a^{2} + 30 a + 21\right)\cdot 59^{4} + \left(31 a^{2} + 57 a + 58\right)\cdot 59^{5} + \left(30 a^{2} + 28 a + 13\right)\cdot 59^{6} + \left(46 a^{2} + 24 a + 15\right)\cdot 59^{7} + \left(2 a^{2} + 36 a + 45\right)\cdot 59^{8} + \left(24 a^{2} + 45 a + 26\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a^{2} + 41 a + 4 + \left(11 a^{2} + 30 a + 25\right)\cdot 59 + \left(29 a^{2} + 47 a + 57\right)\cdot 59^{2} + \left(32 a^{2} + 21 a + 5\right)\cdot 59^{3} + \left(25 a^{2} + 48 a + 35\right)\cdot 59^{4} + \left(50 a^{2} + 25 a + 42\right)\cdot 59^{5} + \left(53 a^{2} + 23 a + 55\right)\cdot 59^{6} + \left(35 a^{2} + 25 a + 46\right)\cdot 59^{7} + \left(48 a^{2} + 20 a + 54\right)\cdot 59^{8} + \left(40 a^{2} + 8 a + 49\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 26 a^{2} + 57 a + 12 + \left(9 a^{2} + 34 a + 21\right)\cdot 59 + \left(40 a^{2} + 31 a + 55\right)\cdot 59^{2} + \left(23 a^{2} + 15 a + 50\right)\cdot 59^{3} + \left(55 a^{2} + 45 a + 11\right)\cdot 59^{4} + \left(23 a^{2} + 27 a + 14\right)\cdot 59^{5} + \left(57 a^{2} + 5\right)\cdot 59^{6} + \left(47 a^{2} + 45 a + 20\right)\cdot 59^{7} + \left(46 a^{2} + 52 a + 54\right)\cdot 59^{8} + \left(4 a^{2} + 21 a + 1\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 22 a^{2} + 24 a + 1 + \left(19 a^{2} + 37 a + 52\right)\cdot 59 + \left(39 a^{2} + 34 a + 51\right)\cdot 59^{2} + \left(26 a^{2} + a + 25\right)\cdot 59^{3} + \left(53 a^{2} + 28 a + 49\right)\cdot 59^{4} + \left(4 a^{2} + 13 a + 8\right)\cdot 59^{5} + \left(27 a^{2} + 18 a + 45\right)\cdot 59^{6} + \left(28 a^{2} + 19 a + 41\right)\cdot 59^{7} + \left(27 a^{2} + 20 a + 23\right)\cdot 59^{8} + \left(55 a^{2} + 29 a\right)\cdot 59^{9} +O(59^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 55 a^{2} + 17 a + 30 + \left(15 a^{2} + 14 a + 3\right)\cdot 59 + \left(27 a^{2} + 46 a + 32\right)\cdot 59^{2} + \left(44 a^{2} + 5 a + 41\right)\cdot 59^{3} + \left(45 a^{2} + 42 a + 38\right)\cdot 59^{4} + \left(3 a^{2} + 32 a + 25\right)\cdot 59^{5} + \left(30 a^{2} + 29 a + 12\right)\cdot 59^{6} + \left(23 a^{2} + 48 a + 17\right)\cdot 59^{7} + \left(9 a^{2} + 28 a + 8\right)\cdot 59^{8} + \left(30 a^{2} + 50 a + 47\right)\cdot 59^{9} +O(59^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $18$ | $2$ | $(1,6)(2,4)(5,9)$ | $2$ |
| $27$ | $2$ | $(1,5)(6,9)$ | $0$ |
| $4$ | $3$ | $(1,5,2)(3,7,8)(4,6,9)$ | $-3 \zeta_{3} - 2$ |
| $4$ | $3$ | $(1,2,5)(3,8,7)(4,9,6)$ | $3 \zeta_{3} + 1$ |
| $6$ | $3$ | $(4,9,6)$ | $-2$ |
| $12$ | $3$ | $(1,5,2)(3,7,8)$ | $1$ |
| $72$ | $3$ | $(1,8,6)(2,7,9)(3,4,5)$ | $1$ |
| $162$ | $4$ | $(1,6,5,9)(2,4)(3,7)$ | $0$ |
| $18$ | $6$ | $(1,6)(2,9)(3,8,7)(4,5)$ | $2 \zeta_{3}$ |
| $18$ | $6$ | $(1,6)(2,9)(3,7,8)(4,5)$ | $-2 \zeta_{3} - 2$ |
| $36$ | $6$ | $(1,4,5,6,2,9)(3,7,8)$ | $-\zeta_{3}$ |
| $36$ | $6$ | $(1,9,2,6,5,4)(3,8,7)$ | $\zeta_{3} + 1$ |
| $36$ | $6$ | $(1,3,5,7,2,8)$ | $-1$ |
| $54$ | $6$ | $(2,5)(3,8)(4,6,9)$ | $0$ |
| $72$ | $9$ | $(1,3,4,5,7,6,2,8,9)$ | $\zeta_{3}$ |
| $72$ | $9$ | $(1,4,7,2,9,3,5,6,8)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.