Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(91079648322445688832\)\(\medspace = 2^{14} \cdot 3^{33} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.650717652052224.3 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1123 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.1.650717652052224.3 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 18x^{6} - 54x^{4} - 126x^{3} - 648x^{2} - 540x - 204 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{3} + 4x + 17 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a^{2} + 7 a + 3 + 4 a^{2} 19 + \left(14 a^{2} + 14\right)\cdot 19^{2} + \left(17 a^{2} + 7 a + 9\right)\cdot 19^{3} + \left(4 a^{2} + 18 a + 7\right)\cdot 19^{4} + \left(14 a + 6\right)\cdot 19^{5} + \left(13 a^{2} + 14 a + 18\right)\cdot 19^{6} + \left(8 a^{2} + 5 a + 8\right)\cdot 19^{7} + \left(a^{2} + 12 a + 15\right)\cdot 19^{8} + \left(14 a^{2} + a + 18\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a^{2} + 5 a + 13 + \left(13 a^{2} + 17 a + 17\right)\cdot 19 + \left(6 a^{2} + 15 a + 12\right)\cdot 19^{2} + \left(2 a + 7\right)\cdot 19^{3} + \left(9 a^{2} + 10 a + 18\right)\cdot 19^{4} + \left(5 a^{2} + 5 a + 13\right)\cdot 19^{5} + \left(16 a^{2} + 2 a + 1\right)\cdot 19^{6} + \left(a^{2} + 6 a + 16\right)\cdot 19^{7} + \left(17 a^{2} + 11 a + 12\right)\cdot 19^{8} + \left(11 a^{2} + 14 a + 6\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a^{2} + 9 a + \left(11 a^{2} + 6 a + 16\right)\cdot 19 + \left(10 a^{2} + 8 a + 16\right)\cdot 19^{2} + \left(14 a^{2} + 10 a + 5\right)\cdot 19^{3} + \left(5 a^{2} + 2 a + 11\right)\cdot 19^{4} + \left(8 a^{2} + 11 a + 10\right)\cdot 19^{5} + \left(16 a + 15\right)\cdot 19^{6} + \left(3 a + 6\right)\cdot 19^{7} + \left(10 a^{2} + 10 a + 17\right)\cdot 19^{8} + \left(17 a^{2} + 14 a + 12\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 13 a^{2} + 10 a + 14 + \left(16 a^{2} + 3 a + 2\right)\cdot 19 + \left(14 a^{2} + 15 a + 18\right)\cdot 19^{2} + \left(2 a^{2} + 6 a + 1\right)\cdot 19^{3} + \left(14 a^{2} + 2 a + 3\right)\cdot 19^{4} + \left(2 a^{2} + 6 a + 13\right)\cdot 19^{5} + \left(15 a^{2} + 7 a + 10\right)\cdot 19^{6} + \left(a^{2} + 12 a + 18\right)\cdot 19^{7} + \left(2 a^{2} + 14 a + 15\right)\cdot 19^{8} + \left(7 a^{2} + 12 a + 1\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a^{2} + 6 a + 18 + \left(18 a^{2} + 14 a + 15\right)\cdot 19 + \left(10 a^{2} + 3 a + 17\right)\cdot 19^{2} + \left(7 a^{2} + 18 a + 18\right)\cdot 19^{3} + \left(17 a^{2} + 9 a + 16\right)\cdot 19^{4} + 18 a^{2} 19^{5} + \left(11 a^{2} + 12 a + 2\right)\cdot 19^{6} + \left(3 a^{2} + 14 a + 10\right)\cdot 19^{7} + \left(9 a^{2} + 15 a + 2\right)\cdot 19^{8} + \left(a^{2} + 12 a + 8\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a^{2} + 3 a + 2 + \left(6 a^{2} + 11 a + 2\right)\cdot 19 + \left(10 a^{2} + 7 a + 6\right)\cdot 19^{2} + \left(3 a^{2} + 18 a + 10\right)\cdot 19^{3} + \left(8 a^{2} + 6 a + 12\right)\cdot 19^{4} + \left(16 a^{2} + 9 a + 11\right)\cdot 19^{5} + \left(9 a^{2} + 15\right)\cdot 19^{6} + \left(10 a^{2} + 8 a + 3\right)\cdot 19^{7} + \left(8 a^{2} + 13 a + 14\right)\cdot 19^{8} + \left(8 a^{2} + a + 11\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 7 a^{2} + 6 a + 17 + \left(14 a^{2} + 4 a + 2\right)\cdot 19 + \left(12 a^{2} + 15 a + 6\right)\cdot 19^{2} + \left(12 a^{2} + 12 a + 9\right)\cdot 19^{3} + \left(15 a^{2} + 9 a + 13\right)\cdot 19^{4} + \left(18 a^{2} + 3 a + 11\right)\cdot 19^{5} + \left(12 a^{2} + 11 a + 17\right)\cdot 19^{6} + \left(6 a^{2} + 17 a + 18\right)\cdot 19^{7} + \left(8 a^{2} + 9 a\right)\cdot 19^{8} + \left(3 a^{2} + 4 a + 11\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 a^{2} + 7 a + 17 + a\cdot 19 + \left(17 a^{2} + 3 a + 15\right)\cdot 19^{2} + \left(9 a + 2\right)\cdot 19^{3} + \left(5 a^{2} + 9 a + 14\right)\cdot 19^{4} + \left(13 a^{2} + 17 a + 15\right)\cdot 19^{5} + \left(8 a^{2} + a + 6\right)\cdot 19^{6} + \left(8 a^{2} + 7 a + 8\right)\cdot 19^{7} + \left(14 a + 6\right)\cdot 19^{8} + \left(12 a^{2} + 2 a + 13\right)\cdot 19^{9} +O(19^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 10 a^{2} + 4 a + 11 + \left(7 a^{2} + 17 a + 17\right)\cdot 19 + \left(16 a^{2} + 6 a + 6\right)\cdot 19^{2} + \left(15 a^{2} + 9 a + 9\right)\cdot 19^{3} + \left(14 a^{2} + 6 a + 16\right)\cdot 19^{4} + \left(10 a^{2} + 7 a + 10\right)\cdot 19^{5} + \left(6 a^{2} + 9 a + 6\right)\cdot 19^{6} + \left(15 a^{2} + 3\right)\cdot 19^{7} + \left(18 a^{2} + 12 a + 9\right)\cdot 19^{8} + \left(18 a^{2} + 10 a + 10\right)\cdot 19^{9} +O(19^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(2,4)(3,6)$ | $0$ |
| $54$ | $2$ | $(1,5)(2,3)(4,6)(8,9)$ | $-2$ |
| $6$ | $3$ | $(3,6,8)$ | $0$ |
| $8$ | $3$ | $(1,5,7)(2,4,9)(3,6,8)$ | $3$ |
| $12$ | $3$ | $(1,5,7)(3,6,8)$ | $-3$ |
| $72$ | $3$ | $(1,2,3)(4,6,5)(7,9,8)$ | $0$ |
| $54$ | $4$ | $(2,3,4,6)(8,9)$ | $0$ |
| $54$ | $6$ | $(1,7,5)(2,4)(3,6)$ | $0$ |
| $108$ | $6$ | $(1,5)(2,3,9,8,4,6)$ | $1$ |
| $72$ | $9$ | $(1,2,3,5,4,6,7,9,8)$ | $0$ |
| $72$ | $9$ | $(1,2,3,7,9,8,5,4,6)$ | $0$ |
| $54$ | $12$ | $(1,5,7)(2,3,4,6)(8,9)$ | $0$ |
| $54$ | $12$ | $(1,7,5)(2,3,4,6)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.