Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(28880462932372000000\)\(\medspace = 2^{8} \cdot 5^{6} \cdot 373^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.83032187200.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T219 |
| Parity: | even |
| Determinant: | 1.373.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.5.83032187200.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{7} - 7x^{6} + 3x^{5} + 14x^{4} - 8x^{3} - 7x^{2} + 7x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{3} + x + 14 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 10 a^{2} + 15 a + 9 + \left(13 a^{2} + 14 a + 7\right)\cdot 17 + \left(16 a^{2} + 10 a + 1\right)\cdot 17^{2} + \left(10 a^{2} + 5 a + 1\right)\cdot 17^{3} + \left(4 a^{2} + 6 a + 7\right)\cdot 17^{4} + \left(15 a + 14\right)\cdot 17^{5} + \left(14 a^{2} + 4 a + 4\right)\cdot 17^{6} + \left(12 a^{2} + 9 a + 4\right)\cdot 17^{7} + \left(5 a^{2} + 2 a + 9\right)\cdot 17^{8} + \left(15 a^{2} + 11\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a^{2} + 2 a + 7 + \left(12 a^{2} + 7 a + 16\right)\cdot 17 + \left(5 a^{2} + 8 a + 1\right)\cdot 17^{2} + \left(8 a^{2} + a + 13\right)\cdot 17^{3} + \left(6 a^{2} + 13 a + 10\right)\cdot 17^{4} + \left(3 a^{2} + 11 a + 9\right)\cdot 17^{5} + \left(6 a^{2} + 14 a + 8\right)\cdot 17^{6} + \left(9 a^{2} + 12 a + 7\right)\cdot 17^{7} + \left(14 a^{2} + 13 a + 14\right)\cdot 17^{8} + 3\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 11 a^{2} + a + 4 + \left(8 a^{2} + 10 a + 4\right)\cdot 17 + \left(12 a^{2} + 3 a + 4\right)\cdot 17^{2} + \left(a^{2} + 3 a + 6\right)\cdot 17^{3} + \left(16 a^{2} + 14 a + 3\right)\cdot 17^{4} + \left(6 a^{2} + 15 a + 13\right)\cdot 17^{5} + \left(9 a^{2} + 3 a + 1\right)\cdot 17^{6} + \left(6 a^{2} + 7 a\right)\cdot 17^{7} + \left(9 a^{2} + 6 a + 6\right)\cdot 17^{8} + \left(14 a^{2} + 16 a + 5\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a^{2} + 14 a + 16 + \left(9 a^{2} + 16\right)\cdot 17 + \left(6 a^{2} + 6 a + 15\right)\cdot 17^{2} + \left(4 a^{2} + 7 a + 12\right)\cdot 17^{3} + \left(14 a^{2} + 7 a + 4\right)\cdot 17^{4} + \left(3 a^{2} + 2 a + 9\right)\cdot 17^{5} + \left(11 a + 11\right)\cdot 17^{6} + \left(10 a^{2} + 3 a + 9\right)\cdot 17^{7} + \left(12 a + 1\right)\cdot 17^{8} + \left(12 a^{2} + 15 a + 9\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a^{2} + a + 11 + \left(11 a^{2} + 9 a\right)\cdot 17 + \left(4 a^{2} + 2 a + 16\right)\cdot 17^{2} + \left(4 a^{2} + 8 a + 7\right)\cdot 17^{3} + \left(13 a^{2} + 13 a + 1\right)\cdot 17^{4} + \left(9 a^{2} + 2 a + 15\right)\cdot 17^{5} + \left(10 a^{2} + 8 a + 13\right)\cdot 17^{6} + \left(14 a^{2} + 16\right)\cdot 17^{7} + \left(a^{2} + 8 a\right)\cdot 17^{8} + \left(4 a^{2} + 4\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a^{2} + 14 a + 6 + \left(4 a^{2} + 16 a + 16\right)\cdot 17 + \left(7 a^{2} + 4 a + 2\right)\cdot 17^{2} + \left(15 a^{2} + 9 a + 12\right)\cdot 17^{3} + \left(12 a + 12\right)\cdot 17^{4} + \left(3 a^{2} + 6 a + 3\right)\cdot 17^{5} + \left(8 a^{2} + 4\right)\cdot 17^{6} + \left(13 a^{2} + 5 a + 10\right)\cdot 17^{7} + \left(4 a^{2} + 16 a + 13\right)\cdot 17^{8} + \left(14 a^{2} + 14 a + 12\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 16 a^{2} + a + 14 + \left(16 a^{2} + 10 a + 7\right)\cdot 17 + \left(3 a^{2} + 3 a + 6\right)\cdot 17^{2} + \left(10 a^{2} + 6 a + 14\right)\cdot 17^{3} + \left(9 a^{2} + 8 a + 12\right)\cdot 17^{4} + \left(10 a^{2} + 15 a + 8\right)\cdot 17^{5} + \left(2 a^{2} + a\right)\cdot 17^{6} + \left(11 a^{2} + 16 a + 3\right)\cdot 17^{7} + \left(14 a^{2} + 3 a + 3\right)\cdot 17^{8} + \left(a^{2} + a + 10\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 3 a^{2} + 5 a + 2 + \left(16 a^{2} + 2 a + 10\right)\cdot 17 + \left(9 a^{2} + a + 12\right)\cdot 17^{2} + \left(7 a^{2} + 2 a + 3\right)\cdot 17^{3} + \left(11 a^{2} + 15 a + 14\right)\cdot 17^{4} + \left(13 a^{2} + 11 a + 15\right)\cdot 17^{5} + \left(11 a^{2} + 11 a + 7\right)\cdot 17^{6} + \left(7 a^{2} + 2 a + 2\right)\cdot 17^{7} + \left(6 a^{2} + 15 a + 11\right)\cdot 17^{8} + \left(4 a^{2} + a + 9\right)\cdot 17^{9} +O(17^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 7 a^{2} + 15 a + 16 + \left(8 a^{2} + 13 a + 4\right)\cdot 17 + \left(9 a + 6\right)\cdot 17^{2} + \left(5 a^{2} + 7 a + 13\right)\cdot 17^{3} + \left(8 a^{2} + 11 a\right)\cdot 17^{4} + \left(16 a^{2} + 2 a + 12\right)\cdot 17^{5} + \left(4 a^{2} + 11 a + 14\right)\cdot 17^{6} + \left(16 a^{2} + 10 a + 13\right)\cdot 17^{7} + \left(9 a^{2} + 6 a + 7\right)\cdot 17^{8} + \left(16 a + 1\right)\cdot 17^{9} +O(17^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $18$ | $2$ | $(1,2)(4,6)(5,8)$ | $2$ |
| $27$ | $2$ | $(3,7)(6,8)$ | $0$ |
| $4$ | $3$ | $(1,6,8)(2,4,5)(3,9,7)$ | $3$ |
| $4$ | $3$ | $(1,8,6)(2,5,4)(3,7,9)$ | $3$ |
| $6$ | $3$ | $(3,7,9)$ | $0$ |
| $12$ | $3$ | $(1,6,8)(2,4,5)$ | $-3$ |
| $72$ | $3$ | $(1,3,4)(2,6,9)(5,8,7)$ | $0$ |
| $162$ | $4$ | $(1,9)(2,4)(3,6,7,8)$ | $0$ |
| $18$ | $6$ | $(1,2)(3,9,7)(4,6)(5,8)$ | $2$ |
| $18$ | $6$ | $(1,2)(3,7,9)(4,6)(5,8)$ | $2$ |
| $36$ | $6$ | $(1,5,8,4,6,2)(3,9,7)$ | $-1$ |
| $36$ | $6$ | $(1,2,6,4,8,5)(3,7,9)$ | $-1$ |
| $36$ | $6$ | $(1,2,6,5,8,4)$ | $-1$ |
| $54$ | $6$ | $(2,4)(3,7,9)(6,8)$ | $0$ |
| $72$ | $9$ | $(1,7,5,6,3,2,8,9,4)$ | $0$ |
| $72$ | $9$ | $(1,5,3,8,4,7,6,2,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.