Basic invariants
| Dimension: | $16$ |
| Group: | $((C_3^2:Q_8):C_3):C_2$ |
| Conductor: | \(146\!\cdots\!000\)\(\medspace = 2^{32} \cdot 3^{20} \cdot 5^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.3.2239488000000.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 24T1334 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_3^2:\GL(2,3)$ |
| Projective stem field: | Galois closure of 9.3.2239488000000.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 6x^{7} - 16x^{6} + 20x^{5} - 36x^{4} + 38x^{3} + 20x^{2} - 27x - 6 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{4} + 3x^{2} + 12x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a^{3} + 9 a^{2} + 3 a + 9 + \left(7 a^{3} + 3 a^{2} + a + 11\right)\cdot 13 + \left(12 a^{3} + 3 a^{2} + 5 a + 9\right)\cdot 13^{2} + \left(9 a^{3} + 8 a^{2} + 4 a + 1\right)\cdot 13^{3} + \left(3 a^{2} + a + 7\right)\cdot 13^{4} + \left(5 a^{3} + 6 a^{2} + 10 a + 8\right)\cdot 13^{5} + \left(2 a^{3} + a^{2} + 10 a + 9\right)\cdot 13^{6} + \left(a^{3} + 5 a^{2} + 2 a + 4\right)\cdot 13^{7} + \left(9 a^{3} + 3 a^{2} + 9 a + 7\right)\cdot 13^{8} + \left(11 a^{3} + 8 a^{2} + 6 a + 11\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 12 + 6\cdot 13 + 2\cdot 13^{2} + 5\cdot 13^{3} + 8\cdot 13^{4} + 6\cdot 13^{5} + 8\cdot 13^{6} + 12\cdot 13^{7} + 8\cdot 13^{8} + 9\cdot 13^{9} +O(13^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 8 a^{3} + 8 a^{2} + 5 a + 7 + \left(5 a^{3} + 4 a^{2} + 11\right)\cdot 13 + \left(3 a^{2} + 5 a + 4\right)\cdot 13^{2} + \left(9 a^{3} + 11 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(5 a^{3} + 3 a^{2} + 11 a + 6\right)\cdot 13^{4} + \left(5 a^{3} + 12 a^{2} + 1\right)\cdot 13^{5} + \left(8 a^{3} + 8 a^{2} + 8 a + 10\right)\cdot 13^{6} + \left(3 a^{3} + 10 a^{2} + 3 a + 2\right)\cdot 13^{7} + \left(10 a^{2} + a + 4\right)\cdot 13^{8} + \left(7 a^{3} + 3 a^{2} + 11 a + 8\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 10 a^{3} + 6 a^{2} + 11 + \left(3 a^{3} + a^{2} + 8\right)\cdot 13 + \left(6 a^{3} + 7 a^{2} + 8 a + 5\right)\cdot 13^{2} + \left(12 a^{3} + 2 a^{2} + 11 a + 10\right)\cdot 13^{3} + \left(4 a^{3} + 4 a^{2} + 2 a + 11\right)\cdot 13^{4} + \left(9 a^{3} + 5 a^{2} + 2 a + 3\right)\cdot 13^{5} + \left(7 a^{3} + 8 a^{2} + 5 a + 6\right)\cdot 13^{6} + \left(7 a^{2} + 11 a + 4\right)\cdot 13^{7} + \left(12 a^{3} + 8 a^{2} + 6 a\right)\cdot 13^{8} + \left(5 a^{3} + 3 a^{2} + 8 a\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 7 a^{3} + 8 a^{2} + 10 a + \left(9 a^{3} + 6 a^{2} + 3 a + 10\right)\cdot 13 + \left(11 a^{3} + 2 a^{2} + 5 a + 8\right)\cdot 13^{2} + \left(7 a^{3} + 3 a^{2} + 12 a + 2\right)\cdot 13^{3} + \left(6 a^{3} + 10 a^{2} + 5 a + 3\right)\cdot 13^{4} + \left(5 a^{3} + 12 a^{2} + 9 a + 6\right)\cdot 13^{5} + \left(5 a^{2} + 12 a + 8\right)\cdot 13^{6} + \left(11 a^{3} + 10 a^{2} + 4 a + 11\right)\cdot 13^{7} + \left(12 a^{3} + 10 a^{2} + 7 a + 10\right)\cdot 13^{8} + \left(4 a^{3} + 3 a^{2} + 4\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 6 }$ |
$=$ |
\( a^{3} + 4 a^{2} + 11 a + 3 + \left(7 a^{3} + 8 a + 5\right)\cdot 13 + \left(5 a^{3} + 6 a^{2} + 2 a + 9\right)\cdot 13^{2} + \left(11 a^{3} + 5 a^{2} + 12 a + 10\right)\cdot 13^{3} + \left(10 a^{2} + 2 a + 11\right)\cdot 13^{4} + \left(6 a^{3} + 4 a^{2} + 6 a + 8\right)\cdot 13^{5} + \left(2 a^{3} + a^{2} + 6 a + 3\right)\cdot 13^{6} + \left(9 a^{3} + 6 a^{2} + 3 a\right)\cdot 13^{7} + \left(9 a^{3} + 11 a^{2} + 12 a + 12\right)\cdot 13^{8} + \left(12 a^{3} + 10 a^{2} + 11\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 7 }$ |
$=$ |
\( a^{3} + 5 a^{2} + 7 a + 11 + \left(6 a^{3} + 4 a^{2} + 2 a + 8\right)\cdot 13 + \left(7 a^{3} + 11\right)\cdot 13^{2} + \left(8 a^{3} + a^{2} + 10 a + 10\right)\cdot 13^{3} + \left(5 a^{3} + 8 a^{2} + 9 a + 5\right)\cdot 13^{4} + \left(9 a^{3} + 2 a^{2} + 8 a + 3\right)\cdot 13^{5} + \left(12 a^{3} + a^{2} + 11\right)\cdot 13^{6} + \left(11 a^{3} + 4 a^{2} + 3 a + 2\right)\cdot 13^{7} + \left(6 a^{3} + 3 a + 9\right)\cdot 13^{8} + \left(7 a^{3} + 3 a^{2} + 7 a + 11\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 5 a^{3} + 7 a^{2} + 7 a + \left(a^{2} + a + 11\right)\cdot 13 + \left(2 a^{3} + 2 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(a^{3} + 6 a^{2} + 3 a + 4\right)\cdot 13^{3} + \left(2 a^{3} + 4 a + 6\right)\cdot 13^{4} + \left(2 a^{3} + 10 a^{2} + 7 a + 4\right)\cdot 13^{5} + \left(10 a^{3} + 11 a^{2} + 8 a + 7\right)\cdot 13^{6} + \left(12 a^{3} + a^{2} + 11 a + 1\right)\cdot 13^{7} + \left(11 a^{3} + 12 a^{2} + 2 a + 5\right)\cdot 13^{8} + \left(11 a^{3} + 7 a^{2} + 12 a + 8\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 4 a^{3} + 5 a^{2} + 9 a + 1 + \left(12 a^{3} + 3 a^{2} + 7 a + 4\right)\cdot 13 + \left(5 a^{3} + a^{2} + 7 a\right)\cdot 13^{2} + \left(4 a^{3} + a^{2} + 11 a + 1\right)\cdot 13^{3} + \left(12 a^{3} + 11 a^{2} + 12 a + 4\right)\cdot 13^{4} + \left(8 a^{3} + 10 a^{2} + 6 a + 8\right)\cdot 13^{5} + \left(7 a^{3} + 12 a^{2} + 12 a + 12\right)\cdot 13^{6} + \left(a^{3} + 5 a^{2} + 10 a + 10\right)\cdot 13^{7} + \left(2 a^{3} + 7 a^{2} + 8 a + 6\right)\cdot 13^{8} + \left(3 a^{3} + 10 a^{2} + 4 a + 11\right)\cdot 13^{9} +O(13^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $16$ |
| $9$ | $2$ | $(1,7)(3,6)(4,5)(8,9)$ | $0$ |
| $36$ | $2$ | $(1,3)(4,5)(6,7)$ | $0$ |
| $8$ | $3$ | $(1,3,8)(2,4,5)(6,7,9)$ | $-2$ |
| $24$ | $3$ | $(1,5,9)(4,8,7)$ | $-2$ |
| $48$ | $3$ | $(1,2,4)(3,8,9)(5,6,7)$ | $1$ |
| $54$ | $4$ | $(1,4,7,5)(3,9,6,8)$ | $0$ |
| $72$ | $6$ | $(1,5,7,6,2,8)(3,9)$ | $0$ |
| $72$ | $6$ | $(1,8,3)(2,7,4,9,5,6)$ | $0$ |
| $54$ | $8$ | $(1,9,4,6,7,8,5,3)$ | $0$ |
| $54$ | $8$ | $(1,8,4,3,7,9,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.