Basic invariants
| Dimension: | $6$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(630247042161\)\(\medspace = 3^{16} \cdot 11^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.13922730113193.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T220 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.5.13922730113193.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 6x^{6} - 27x^{5} + 45x^{4} + 30x^{3} - 108x^{2} + 45x + 25 \)
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The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{3} + 3x + 81 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 59 a^{2} + 59 a + 35 + \left(9 a^{2} + 62 a + 19\right)\cdot 83 + \left(31 a^{2} + 6 a + 62\right)\cdot 83^{2} + \left(16 a^{2} + 21 a + 32\right)\cdot 83^{3} + \left(36 a^{2} + 68 a + 72\right)\cdot 83^{4} + \left(42 a^{2} + 21 a + 1\right)\cdot 83^{5} + \left(51 a^{2} + 17 a + 20\right)\cdot 83^{6} + \left(64 a^{2} + 75 a + 46\right)\cdot 83^{7} + \left(69 a^{2} + 74 a + 56\right)\cdot 83^{8} + \left(32 a^{2} + 37 a + 65\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 + 65\cdot 83 + 24\cdot 83^{2} + 50\cdot 83^{3} + 57\cdot 83^{4} + 58\cdot 83^{5} + 69\cdot 83^{6} + 61\cdot 83^{7} + 64\cdot 83^{8} + 21\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 71 + 44\cdot 83 + 12\cdot 83^{2} + 81\cdot 83^{3} + 69\cdot 83^{4} + 15\cdot 83^{5} + 62\cdot 83^{6} + 22\cdot 83^{7} + 81\cdot 83^{8} + 33\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 a^{2} + 19 a + 24 + \left(53 a^{2} + 25 a + 23\right)\cdot 83 + \left(80 a^{2} + 24 a + 78\right)\cdot 83^{2} + \left(74 a^{2} + 14 a + 66\right)\cdot 83^{3} + \left(11 a^{2} + 17 a + 23\right)\cdot 83^{4} + \left(50 a^{2} + 27 a + 17\right)\cdot 83^{5} + \left(22 a^{2} + 27 a + 45\right)\cdot 83^{6} + \left(35 a^{2} + 76 a + 70\right)\cdot 83^{7} + \left(5 a^{2} + 77 a + 10\right)\cdot 83^{8} + \left(22 a^{2} + 70 a + 44\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 67 + 55\cdot 83 + 45\cdot 83^{2} + 34\cdot 83^{3} + 38\cdot 83^{4} + 8\cdot 83^{5} + 34\cdot 83^{6} + 81\cdot 83^{7} + 19\cdot 83^{8} + 27\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 76 a + \left(22 a^{2} + 75 a + 44\right)\cdot 83 + \left(14 a^{2} + 3 a + 28\right)\cdot 83^{2} + \left(27 a^{2} + 46 a + 54\right)\cdot 83^{3} + \left(40 a^{2} + 40 a + 80\right)\cdot 83^{4} + \left(70 a^{2} + 2 a + 57\right)\cdot 83^{5} + \left(42 a^{2} + 35 a + 2\right)\cdot 83^{6} + \left(46 a^{2} + 51 a + 10\right)\cdot 83^{7} + \left(41 a^{2} + 61 a\right)\cdot 83^{8} + \left(9 a^{2} + 66 a + 19\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 18 a^{2} + 63 a + 36 + \left(54 a^{2} + 17 a + 25\right)\cdot 83 + \left(25 a^{2} + 49 a + 51\right)\cdot 83^{2} + \left(81 a^{2} + 10 a + 79\right)\cdot 83^{3} + \left(4 a^{2} + 25 a + 9\right)\cdot 83^{4} + \left(74 a^{2} + 15 a + 65\right)\cdot 83^{5} + \left(44 a^{2} + 79 a + 6\right)\cdot 83^{6} + \left(60 a^{2} + 77 a + 38\right)\cdot 83^{7} + \left(66 a^{2} + 79 a + 50\right)\cdot 83^{8} + \left(81 a^{2} + 79 a + 80\right)\cdot 83^{9} +O(83^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 65 a^{2} + 27 a + 47 + \left(6 a^{2} + 72 a + 13\right)\cdot 83 + \left(43 a^{2} + 29 a + 3\right)\cdot 83^{2} + \left(57 a^{2} + 26 a + 32\right)\cdot 83^{3} + \left(37 a^{2} + 17 a + 75\right)\cdot 83^{4} + \left(21 a^{2} + 65 a + 42\right)\cdot 83^{5} + \left(78 a^{2} + 51 a + 73\right)\cdot 83^{6} + \left(58 a^{2} + 36 a + 34\right)\cdot 83^{7} + \left(57 a^{2} + 24 a + 32\right)\cdot 83^{8} + \left(74 a^{2} + 19 a + 66\right)\cdot 83^{9} +O(83^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 12 a^{2} + 5 a + 24 + \left(20 a^{2} + 78 a + 40\right)\cdot 83 + \left(54 a^{2} + 51 a + 25\right)\cdot 83^{2} + \left(74 a^{2} + 47 a + 66\right)\cdot 83^{3} + \left(34 a^{2} + 80 a + 69\right)\cdot 83^{4} + \left(73 a^{2} + 33 a + 63\right)\cdot 83^{5} + \left(8 a^{2} + 38 a + 17\right)\cdot 83^{6} + \left(66 a^{2} + 14 a + 49\right)\cdot 83^{7} + \left(7 a^{2} + 13 a + 15\right)\cdot 83^{8} + \left(28 a^{2} + 57 a + 56\right)\cdot 83^{9} +O(83^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $18$ | $2$ | $(1,6)(4,8)(7,9)$ | $-2$ |
| $27$ | $2$ | $(2,5)(6,7)$ | $2$ |
| $4$ | $3$ | $(1,9,4)(2,3,5)(6,8,7)$ | $-3$ |
| $4$ | $3$ | $(1,4,9)(2,5,3)(6,7,8)$ | $-3$ |
| $6$ | $3$ | $(6,8,7)$ | $3$ |
| $12$ | $3$ | $(1,9,4)(6,7,8)$ | $0$ |
| $72$ | $3$ | $(1,8,5)(2,4,7)(3,9,6)$ | $0$ |
| $162$ | $4$ | $(1,8,9,6)(2,5)(4,7)$ | $0$ |
| $18$ | $6$ | $(1,8)(2,5,3)(4,6)(7,9)$ | $1$ |
| $18$ | $6$ | $(1,8)(2,3,5)(4,6)(7,9)$ | $1$ |
| $36$ | $6$ | $(1,8,9,6,4,7)$ | $-2$ |
| $36$ | $6$ | $(1,3,9,2,4,5)(6,7,8)$ | $1$ |
| $36$ | $6$ | $(1,5,4,2,9,3)(6,8,7)$ | $1$ |
| $54$ | $6$ | $(2,5,3)(4,9)(7,8)$ | $-1$ |
| $72$ | $9$ | $(1,5,6,9,2,8,4,3,7)$ | $0$ |
| $72$ | $9$ | $(1,6,2,4,7,5,9,8,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.