Basic invariants
| Dimension: | $3$ |
| Group: | $(C_3^2:C_3):C_2$ |
| Conductor: | \(210681\)\(\medspace = 3^{6} \cdot 17^{2} \) |
| Artin stem field: | Galois closure of 9.3.1198435061547.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | 18T24 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_3:S_3$ |
| Projective stem field: | Galois closure of 9.1.1039043198361249.5 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 6x^{7} + 5x^{6} - 39x^{5} + 96x^{4} - 17x^{3} - 90x^{2} + 24 \)
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The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{3} + 7x + 59 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 18 a^{2} + 49 a + 42 + \left(25 a^{2} + 54 a + 7\right)\cdot 61 + \left(34 a^{2} + 44 a + 22\right)\cdot 61^{2} + \left(26 a^{2} + 58 a + 15\right)\cdot 61^{3} + \left(42 a^{2} + 54 a + 4\right)\cdot 61^{4} + \left(51 a^{2} + 39 a + 43\right)\cdot 61^{5} + \left(41 a^{2} + 18 a + 42\right)\cdot 61^{6} + \left(a^{2} + 5 a + 21\right)\cdot 61^{7} + \left(44 a^{2} + 21 a + 16\right)\cdot 61^{8} + \left(2 a^{2} + 13 a + 45\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 42 a^{2} + 30 a + 32 + \left(28 a^{2} + a + 23\right)\cdot 61 + \left(17 a^{2} + 2 a + 24\right)\cdot 61^{2} + \left(35 a^{2} + 56 a + 15\right)\cdot 61^{3} + \left(60 a^{2} + 29 a + 48\right)\cdot 61^{4} + \left(46 a^{2} + 51 a\right)\cdot 61^{5} + \left(28 a^{2} + 39 a + 2\right)\cdot 61^{6} + \left(36 a^{2} + 46 a + 1\right)\cdot 61^{7} + \left(40 a^{2} + 40 a + 41\right)\cdot 61^{8} + \left(16 a^{2} + 56 a + 8\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 15\cdot 61 + 41\cdot 61^{2} + 57\cdot 61^{3} + 53\cdot 61^{4} + 56\cdot 61^{5} + 5\cdot 61^{6} + 48\cdot 61^{7} + 26\cdot 61^{8} + 5\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 59 a^{2} + 51 a + 10 + \left(22 a^{2} + 59 a + 10\right)\cdot 61 + \left(54 a^{2} + 46 a + 56\right)\cdot 61^{2} + \left(8 a^{2} + 3 a + 39\right)\cdot 61^{3} + \left(18 a^{2} + 24 a + 10\right)\cdot 61^{4} + \left(2 a^{2} + 11 a + 24\right)\cdot 61^{5} + \left(55 a^{2} + 25 a + 42\right)\cdot 61^{6} + \left(20 a^{2} + 33 a + 3\right)\cdot 61^{7} + \left(2 a^{2} + 34 a + 36\right)\cdot 61^{8} + \left(15 a^{2} + 31 a + 16\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 55 + 53\cdot 61 + 58\cdot 61^{2} + 20\cdot 61^{3} + 60\cdot 61^{4} + 32\cdot 61^{5} + 45\cdot 61^{6} + 30\cdot 61^{7} + 7\cdot 61^{8} + 41\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 22 a^{2} + 37 a + \left(52 a^{2} + 27 a + 5\right)\cdot 61 + \left(44 a^{2} + 27 a + 32\right)\cdot 61^{2} + \left(20 a^{2} + 24 a + 54\right)\cdot 61^{3} + \left(60 a^{2} + 31 a + 44\right)\cdot 61^{4} + \left(10 a^{2} + 12 a + 3\right)\cdot 61^{5} + \left(24 a^{2} + 40 a\right)\cdot 61^{6} + \left(59 a^{2} + 38 a + 41\right)\cdot 61^{7} + \left(36 a^{2} + 9 a + 14\right)\cdot 61^{8} + \left(44 a^{2} + 56 a + 12\right)\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 + 4\cdot 61 + 54\cdot 61^{2} + 7\cdot 61^{3} + 18\cdot 61^{4} + 37\cdot 61^{5} + 12\cdot 61^{6} + 39\cdot 61^{7} + 30\cdot 61^{8} + 16\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 41 a^{2} + 34 a + 48 + \left(46 a^{2} + 34 a + 59\right)\cdot 61 + \left(22 a^{2} + 47 a + 50\right)\cdot 61^{2} + \left(31 a^{2} + 32 a + 22\right)\cdot 61^{3} + \left(43 a^{2} + 5 a + 27\right)\cdot 61^{4} + \left(47 a^{2} + 37 a + 12\right)\cdot 61^{5} + \left(42 a^{2} + 56 a + 46\right)\cdot 61^{6} + \left(41 a^{2} + 49 a + 39\right)\cdot 61^{7} + \left(21 a^{2} + 16 a + 24\right)\cdot 61^{8} + \left(a^{2} + 34 a + 54\right)\cdot 61^{9} +O(61^{10})\)
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| $r_{ 9 }$ | $=$ |
\( a^{2} + 43 a + 44 + \left(7 a^{2} + 4 a + 3\right)\cdot 61 + \left(9 a^{2} + 14 a + 26\right)\cdot 61^{2} + \left(60 a^{2} + 7 a + 9\right)\cdot 61^{3} + \left(18 a^{2} + 37 a + 37\right)\cdot 61^{4} + \left(23 a^{2} + 30 a + 32\right)\cdot 61^{5} + \left(51 a^{2} + 2 a + 46\right)\cdot 61^{6} + \left(22 a^{2} + 9 a + 18\right)\cdot 61^{7} + \left(37 a^{2} + 60 a + 46\right)\cdot 61^{8} + \left(41 a^{2} + 51 a + 43\right)\cdot 61^{9} +O(61^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,3)(2,7)(5,9)$ | $-1$ |
| $1$ | $3$ | $(1,2,9)(3,7,5)(4,8,6)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,9,2)(3,5,7)(4,6,8)$ | $-3 \zeta_{3} - 3$ |
| $6$ | $3$ | $(1,6,5)(2,4,3)(7,9,8)$ | $0$ |
| $6$ | $3$ | $(1,4,5)(2,8,3)(6,7,9)$ | $0$ |
| $6$ | $3$ | $(1,8,5)(2,6,3)(4,7,9)$ | $0$ |
| $6$ | $3$ | $(3,5,7)(4,8,6)$ | $0$ |
| $9$ | $6$ | $(1,7,9,3,2,5)(4,8,6)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,5,2,3,9,7)(4,6,8)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.