Basic invariants
| Dimension: | $3$ |
| Group: | $(C_3^2:C_3):C_2$ |
| Conductor: | \(184900\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 43^{2} \) |
| Artin stem field: | Galois closure of 9.3.3180280000.1 |
| 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 9.1.3418801000000.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 3x^{7} - 4x^{6} - 2x^{5} + 11x^{4} + 5x^{3} - 5x^{2} - 2x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$:
\( x^{3} + 3x + 42 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 15\cdot 47 + 2\cdot 47^{2} + 27\cdot 47^{3} + 22\cdot 47^{4} + 5\cdot 47^{5} + 13\cdot 47^{6} + 33\cdot 47^{7} + 6\cdot 47^{8} + 12\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 31 + 36\cdot 47 + 7\cdot 47^{2} + 32\cdot 47^{3} + 8\cdot 47^{4} + 9\cdot 47^{5} + 5\cdot 47^{6} + 29\cdot 47^{7} + 19\cdot 47^{8} + 10\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 a^{2} + 15 a + 38 + \left(23 a^{2} + 40 a + 15\right)\cdot 47 + \left(19 a^{2} + 10 a + 18\right)\cdot 47^{2} + \left(42 a^{2} + 33 a + 13\right)\cdot 47^{3} + \left(7 a^{2} + 9 a + 13\right)\cdot 47^{4} + \left(10 a^{2} + 43 a + 6\right)\cdot 47^{5} + \left(30 a^{2} + 33 a + 17\right)\cdot 47^{6} + \left(45 a^{2} + 11 a + 23\right)\cdot 47^{7} + \left(13 a^{2} + 20 a + 4\right)\cdot 47^{8} + \left(23 a^{2} + 42 a + 19\right)\cdot 47^{9} +O(47^{10})\)
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| $r_{ 4 }$ | $=$ |
\( 9 a^{2} + 7 a + 29 + \left(40 a^{2} + 18 a + 1\right)\cdot 47 + \left(40 a^{2} + 3 a + 14\right)\cdot 47^{2} + \left(40 a^{2} + 33 a + 10\right)\cdot 47^{3} + \left(36 a + 46\right)\cdot 47^{4} + \left(32 a^{2} + 17 a + 2\right)\cdot 47^{5} + \left(32 a^{2} + 3 a + 22\right)\cdot 47^{6} + \left(7 a^{2} + 17 a + 41\right)\cdot 47^{7} + \left(41 a^{2} + 28 a + 11\right)\cdot 47^{8} + \left(29 a^{2} + 12 a + 32\right)\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 19 a^{2} + 9 a + 45 + \left(29 a^{2} + 15 a + 2\right)\cdot 47 + \left(11 a + 1\right)\cdot 47^{2} + \left(17 a^{2} + 43 a + 32\right)\cdot 47^{3} + \left(16 a^{2} + 3 a + 31\right)\cdot 47^{4} + \left(7 a^{2} + 16 a + 31\right)\cdot 47^{5} + \left(39 a^{2} + 44 a + 11\right)\cdot 47^{6} + \left(20 a^{2} + 40 a + 16\right)\cdot 47^{7} + \left(18 a^{2} + 16 a + 12\right)\cdot 47^{8} + \left(42 a^{2} + 17 a + 19\right)\cdot 47^{9} +O(47^{10})\)
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| $r_{ 6 }$ | $=$ |
\( a^{2} + 25 a + 13 + \left(30 a^{2} + 35 a + 28\right)\cdot 47 + \left(33 a^{2} + 32 a + 46\right)\cdot 47^{2} + \left(10 a^{2} + 27 a + 43\right)\cdot 47^{3} + \left(38 a^{2} + 26\right)\cdot 47^{4} + \left(4 a^{2} + 33 a + 42\right)\cdot 47^{5} + \left(31 a^{2} + 9 a + 18\right)\cdot 47^{6} + \left(40 a^{2} + 18 a + 13\right)\cdot 47^{7} + \left(38 a^{2} + 45 a + 7\right)\cdot 47^{8} + \left(40 a^{2} + 38 a + 7\right)\cdot 47^{9} +O(47^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 36 a^{2} + 2 a + 32 + \left(35 a^{2} + 22 a + 15\right)\cdot 47 + \left(36 a^{2} + 19 a + 26\right)\cdot 47^{2} + \left(17 a^{2} + 2 a + 33\right)\cdot 47^{3} + \left(13 a^{2} + 16 a + 25\right)\cdot 47^{4} + \left(45 a^{2} + 8 a + 13\right)\cdot 47^{5} + \left(33 a^{2} + 42 a + 1\right)\cdot 47^{6} + \left(31 a^{2} + 38\right)\cdot 47^{7} + \left(22 a^{2} + 40 a + 20\right)\cdot 47^{8} + \left(3 a^{2} + 26 a + 35\right)\cdot 47^{9} +O(47^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 39 a^{2} + 36 a + 38 + \left(28 a^{2} + 9 a + 1\right)\cdot 47 + \left(9 a^{2} + 16 a + 19\right)\cdot 47^{2} + \left(12 a^{2} + a + 22\right)\cdot 47^{3} + \left(17 a^{2} + 27 a + 33\right)\cdot 47^{4} + \left(41 a^{2} + 22 a + 5\right)\cdot 47^{5} + \left(20 a^{2} + 7 a + 22\right)\cdot 47^{6} + \left(41 a^{2} + 5 a + 10\right)\cdot 47^{7} + \left(5 a^{2} + 37 a + 34\right)\cdot 47^{8} + \left(a^{2} + 2 a + 30\right)\cdot 47^{9} +O(47^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 7 + 23\cdot 47 + 5\cdot 47^{2} + 20\cdot 47^{3} + 26\cdot 47^{4} + 23\cdot 47^{5} + 29\cdot 47^{6} + 29\cdot 47^{7} + 23\cdot 47^{8} + 21\cdot 47^{9} +O(47^{10})\)
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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 |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(3,5)(4,8)(6,7)$ | $-1$ |
| $1$ | $3$ | $(1,2,9)(3,4,6)(5,8,7)$ | $-3 \zeta_{3} - 3$ |
| $1$ | $3$ | $(1,9,2)(3,6,4)(5,7,8)$ | $3 \zeta_{3}$ |
| $6$ | $3$ | $(1,7,6)(2,5,3)(4,9,8)$ | $0$ |
| $6$ | $3$ | $(1,8,6)(2,7,3)(4,9,5)$ | $0$ |
| $6$ | $3$ | $(3,4,6)(5,7,8)$ | $0$ |
| $6$ | $3$ | $(1,6,5)(2,3,8)(4,7,9)$ | $0$ |
| $9$ | $6$ | $(1,2,9)(3,8,6,5,4,7)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,9,2)(3,7,4,5,6,8)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.