Basic invariants
| Dimension: | $12$ |
| Group: | $S_3 \wr C_3 $ |
| Conductor: | \(270945468811850625\)\(\medspace = 3^{12} \cdot 5^{4} \cdot 13^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.12825821008845.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T206 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr C_3$ |
| Projective stem field: | Galois closure of 9.5.12825821008845.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{7} - 17x^{6} - 36x^{5} + 21x^{4} + 39x^{3} + 9x^{2} + 90x + 53 \)
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The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{3} + 2x + 27 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 20 a^{2} + 9 a + 23 + \left(5 a + 24\right)\cdot 29 + \left(25 a^{2} + 24 a + 9\right)\cdot 29^{2} + \left(11 a^{2} + 25 a + 1\right)\cdot 29^{3} + \left(25 a^{2} + 18 a + 19\right)\cdot 29^{4} + \left(26 a^{2} + 2 a + 7\right)\cdot 29^{5} + \left(a^{2} + 4\right)\cdot 29^{6} + \left(14 a^{2} + 3 a + 19\right)\cdot 29^{7} + \left(28 a^{2} + a + 11\right)\cdot 29^{8} + \left(16 a^{2} + 19 a + 10\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 a^{2} + 26 + \left(2 a^{2} + 6 a + 23\right)\cdot 29 + \left(25 a^{2} + 8 a + 19\right)\cdot 29^{2} + \left(13 a^{2} + 7 a + 1\right)\cdot 29^{3} + \left(7 a^{2} + 16 a + 26\right)\cdot 29^{4} + \left(23 a^{2} + 23 a + 28\right)\cdot 29^{5} + \left(13 a^{2} + 28 a + 5\right)\cdot 29^{6} + \left(a^{2} + 21 a + 27\right)\cdot 29^{7} + \left(15 a^{2} + 8 a + 4\right)\cdot 29^{8} + \left(6 a^{2} + 21 a + 10\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a^{2} + 15 a + 28 + \left(8 a^{2} + 28 a + 4\right)\cdot 29 + \left(6 a^{2} + 26 a + 26\right)\cdot 29^{2} + \left(15 a^{2} + 27 a + 12\right)\cdot 29^{3} + \left(27 a^{2} + 19 a + 6\right)\cdot 29^{4} + \left(5 a^{2} + 3 a + 9\right)\cdot 29^{5} + \left(27 a^{2} + 6 a + 8\right)\cdot 29^{6} + \left(12 a^{2} + 21 a + 1\right)\cdot 29^{7} + \left(27 a^{2} + 6 a + 20\right)\cdot 29^{8} + \left(24 a^{2} + 26 a + 24\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a^{2} + 23 a + 10 + \left(24 a^{2} + 11 a + 26\right)\cdot 29 + \left(14 a^{2} + 5 a + 27\right)\cdot 29^{2} + \left(10 a^{2} + 5 a + 25\right)\cdot 29^{3} + \left(5 a^{2} + 15 a + 5\right)\cdot 29^{4} + \left(15 a^{2} + 22 a + 2\right)\cdot 29^{5} + \left(17 a^{2} + 22 a + 5\right)\cdot 29^{6} + \left(2 a^{2} + 3 a + 26\right)\cdot 29^{7} + \left(16 a^{2} + 3 a + 4\right)\cdot 29^{8} + \left(27 a^{2} + 14 a + 28\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 17 + \left(20 a^{2} + a + 8\right)\cdot 29 + \left(24 a^{2} + 8 a + 19\right)\cdot 29^{2} + \left(22 a^{2} + 6 a + 13\right)\cdot 29^{3} + \left(22 a^{2} + 13 a + 17\right)\cdot 29^{4} + \left(5 a^{2} + 3 a + 5\right)\cdot 29^{5} + \left(20 a^{2} + 9 a + 24\right)\cdot 29^{6} + \left(14 a^{2} + 15\right)\cdot 29^{7} + \left(24 a^{2} + 12 a + 17\right)\cdot 29^{8} + \left(4 a^{2} + 6 a + 17\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 15 a^{2} + 16 a + 8 + \left(6 a^{2} + 21 a\right)\cdot 29 + \left(8 a^{2} + 12 a + 7\right)\cdot 29^{2} + \left(21 a^{2} + 15 a + 21\right)\cdot 29^{3} + \left(27 a^{2} + 28 a + 4\right)\cdot 29^{4} + \left(28 a^{2} + a + 17\right)\cdot 29^{5} + \left(23 a^{2} + 20 a + 19\right)\cdot 29^{6} + \left(12 a^{2} + 6 a + 3\right)\cdot 29^{7} + \left(18 a^{2} + 8 a + 19\right)\cdot 29^{8} + \left(17 a^{2} + a + 5\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 12 a^{2} + 27 a + 22 + \left(14 a^{2} + 7 a + 23\right)\cdot 29 + \left(14 a^{2} + 18 a + 24\right)\cdot 29^{2} + \left(21 a^{2} + 14 a + 23\right)\cdot 29^{3} + \left(2 a^{2} + 9 a + 17\right)\cdot 29^{4} + \left(23 a^{2} + 23 a + 2\right)\cdot 29^{5} + \left(6 a^{2} + 2 a + 1\right)\cdot 29^{6} + \left(3 a^{2} + a + 24\right)\cdot 29^{7} + \left(12 a^{2} + 14 a + 18\right)\cdot 29^{8} + \left(15 a^{2} + a + 27\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 26 a^{2} + 22 a + 2 + \left(13 a^{2} + 15 a + 23\right)\cdot 29 + \left(18 a^{2} + 15 a + 10\right)\cdot 29^{2} + \left(24 a^{2} + 17 a + 18\right)\cdot 29^{3} + 5\cdot 29^{4} + \left(8 a^{2} + 3 a + 21\right)\cdot 29^{5} + \left(20 a^{2} + 26 a + 28\right)\cdot 29^{6} + \left(11 a^{2} + 24 a + 15\right)\cdot 29^{7} + \left(17 a^{2} + 13 a + 6\right)\cdot 29^{8} + \left(25 a^{2} + 8 a + 12\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 24 a^{2} + 20 a + 9 + \left(25 a^{2} + 17 a + 9\right)\cdot 29 + \left(7 a^{2} + 25 a + 28\right)\cdot 29^{2} + \left(3 a^{2} + 24 a + 25\right)\cdot 29^{3} + \left(25 a^{2} + 22 a + 12\right)\cdot 29^{4} + \left(7 a^{2} + 2 a + 21\right)\cdot 29^{5} + \left(13 a^{2} + 18\right)\cdot 29^{6} + \left(13 a^{2} + 4 a + 11\right)\cdot 29^{7} + \left(14 a^{2} + 19 a + 12\right)\cdot 29^{8} + \left(5 a^{2} + 17 a + 8\right)\cdot 29^{9} +O(29^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(3,6)$ | $4$ |
| $27$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $27$ | $2$ | $(2,9)(3,6)$ | $0$ |
| $6$ | $3$ | $(4,5,8)$ | $0$ |
| $8$ | $3$ | $(1,2,9)(3,6,7)(4,5,8)$ | $3$ |
| $12$ | $3$ | $(1,2,9)(4,5,8)$ | $-3$ |
| $36$ | $3$ | $(1,3,4)(2,6,5)(7,8,9)$ | $0$ |
| $36$ | $3$ | $(1,4,3)(2,5,6)(7,9,8)$ | $0$ |
| $18$ | $6$ | $(3,6)(4,5,8)$ | $-2$ |
| $18$ | $6$ | $(1,2,9)(3,6)$ | $-2$ |
| $36$ | $6$ | $(1,2,9)(3,6)(4,5,8)$ | $1$ |
| $54$ | $6$ | $(2,9)(3,6)(4,5,8)$ | $0$ |
| $108$ | $6$ | $(1,3,5,2,6,4)(7,8,9)$ | $0$ |
| $108$ | $6$ | $(1,4,6,2,5,3)(7,9,8)$ | $0$ |
| $72$ | $9$ | $(1,3,4,2,6,5,9,7,8)$ | $0$ |
| $72$ | $9$ | $(1,4,6,9,8,3,2,5,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.