Basic invariants
Dimension: | $8$ |
Group: | $((C_3^2:Q_8):C_3):C_2$ |
Conductor: | \(36889110963\)\(\medspace = 3^{9} \cdot 37^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.3.36889110963.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 18T157 |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $C_3^2:\GL(2,3)$ |
Projective stem field: | Galois closure of 9.3.36889110963.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} + 3x^{7} - 3x^{6} + 6x^{3} + 3x^{2} - 9x - 5 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{4} + 2x^{2} + 11x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 11 a^{3} + 4 a^{2} + 2 a + 8 + \left(3 a^{3} + 4 a^{2} + 14 a + 10\right)\cdot 19 + \left(8 a^{3} + 18 a^{2} + 17 a + 17\right)\cdot 19^{2} + \left(3 a^{3} + 7 a^{2} + 17 a + 12\right)\cdot 19^{3} + \left(11 a^{3} + 5 a^{2} + 8 a + 3\right)\cdot 19^{4} + \left(12 a^{3} + 9 a^{2} + 4 a + 18\right)\cdot 19^{5} + \left(17 a^{3} + 10 a^{2} + 14\right)\cdot 19^{6} + \left(11 a^{3} + 6 a^{2} + 6 a + 13\right)\cdot 19^{7} + \left(8 a^{3} + a^{2} + a + 18\right)\cdot 19^{8} + \left(18 a^{3} + 16 a^{2} + 13 a + 5\right)\cdot 19^{9} +O(19^{10})\)
$r_{ 2 }$ |
$=$ |
\( 2 + 11\cdot 19 + 15\cdot 19^{2} + 2\cdot 19^{3} + 6\cdot 19^{4} + 16\cdot 19^{5} + 17\cdot 19^{6} + 19^{8} + 12\cdot 19^{9} +O(19^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 4 a^{3} + 18 a^{2} + 11 a + 7 + \left(8 a^{3} + 4 a^{2} + 8 a + 11\right)\cdot 19 + \left(5 a^{3} + 15 a^{2} + 16 a + 15\right)\cdot 19^{2} + \left(a^{3} + 6 a^{2} + 17 a + 17\right)\cdot 19^{3} + \left(9 a^{3} + 16 a^{2} + 18 a + 1\right)\cdot 19^{4} + \left(12 a^{3} + 3 a^{2} + 11 a + 7\right)\cdot 19^{5} + \left(15 a^{3} + 4 a^{2} + 18 a + 6\right)\cdot 19^{6} + \left(15 a^{3} + 6 a^{2} + 8 a + 12\right)\cdot 19^{7} + \left(13 a^{3} + a^{2} + 6 a + 4\right)\cdot 19^{8} + \left(2 a^{3} + 16 a^{2} + 3 a + 9\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 13 a^{3} + 13 a^{2} + 5 a + 7 + \left(8 a^{3} + 2 a^{2} + 14 a + 9\right)\cdot 19 + \left(3 a^{3} + 6 a^{2} + 16 a + 13\right)\cdot 19^{2} + \left(8 a^{2} + 14 a + 8\right)\cdot 19^{3} + \left(15 a^{3} + 2 a^{2} + 13 a + 14\right)\cdot 19^{4} + \left(17 a^{3} + 10 a^{2} + 15 a\right)\cdot 19^{5} + \left(12 a^{3} + 4 a^{2} + 5 a + 1\right)\cdot 19^{6} + \left(12 a^{3} + a^{2} + 5 a + 2\right)\cdot 19^{7} + \left(15 a^{3} + 9 a^{2} + 18 a + 16\right)\cdot 19^{8} + \left(17 a^{3} + 7 a^{2} + 4 a + 9\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 16 a^{3} + 8 a^{2} + 17 a + 3 + \left(16 a^{3} + 10 a + 3\right)\cdot 19 + \left(10 a^{3} + 7 a^{2} + a + 9\right)\cdot 19^{2} + \left(14 a^{3} + 13 a^{2} + 6 a + 18\right)\cdot 19^{3} + \left(15 a^{3} + 8 a^{2} + 7\right)\cdot 19^{4} + \left(2 a^{3} + 14 a^{2}\right)\cdot 19^{5} + \left(14 a^{3} + 12 a^{2} + 3 a + 5\right)\cdot 19^{6} + \left(a^{3} + 12 a^{2} + 18\right)\cdot 19^{7} + \left(12 a^{3} + 5 a^{2} + 18 a + 6\right)\cdot 19^{8} + \left(16 a^{3} + 14 a^{2} + 11 a + 16\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 18 a^{3} + 16 a^{2} + 13 a + 18 + \left(4 a^{3} + 13 a^{2} + 3\right)\cdot 19 + \left(17 a^{3} + 6 a^{2} + 2 a + 9\right)\cdot 19^{2} + \left(17 a^{3} + 15 a^{2} + a\right)\cdot 19^{3} + \left(4 a^{3} + 18 a^{2} + 13 a\right)\cdot 19^{4} + \left(7 a^{3} + 16 a^{2} + 11\right)\cdot 19^{5} + \left(10 a^{3} + 4 a^{2} + 16 a + 13\right)\cdot 19^{6} + \left(3 a^{3} + 9 a^{2} + 17 a + 10\right)\cdot 19^{7} + \left(6 a^{3} + 8 a\right)\cdot 19^{8} + \left(17 a^{2} + 17 a + 8\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 7 a^{3} + 11 a^{2} + 11 a + 1 + \left(5 a^{3} + 12 a^{2} + 3 a + 5\right)\cdot 19 + \left(8 a^{3} + 10 a^{2} + 5 a + 1\right)\cdot 19^{2} + \left(6 a^{3} + 5 a^{2} + 3 a + 2\right)\cdot 19^{3} + \left(3 a^{3} + 4 a^{2} + 13 a + 9\right)\cdot 19^{4} + \left(4 a^{3} + 14 a^{2} + 16 a + 15\right)\cdot 19^{5} + \left(10 a^{3} + a^{2} + 4 a + 1\right)\cdot 19^{6} + \left(2 a^{3} + 8 a^{2} + 6 a + 9\right)\cdot 19^{7} + \left(12 a^{3} + 9 a + 13\right)\cdot 19^{8} + \left(10 a^{3} + 18 a^{2} + 7 a + 14\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 10 a^{3} + a^{2} + 3 a + 13 + \left(7 a^{3} + 2 a^{2} + 12 a + 8\right)\cdot 19 + \left(6 a^{3} + 18 a^{2} + 17 a + 16\right)\cdot 19^{2} + \left(5 a^{3} + 15 a + 5\right)\cdot 19^{3} + \left(2 a^{3} + 8 a^{2} + 10 a + 5\right)\cdot 19^{4} + \left(10 a^{3} + 15 a^{2} + 2 a + 9\right)\cdot 19^{5} + \left(15 a^{2} + 13 a + 14\right)\cdot 19^{6} + \left(a^{3} + 14 a^{2} + 14 a + 9\right)\cdot 19^{7} + \left(4 a^{3} + 3 a^{2} + 11 a\right)\cdot 19^{8} + \left(3 a^{3} + 18 a^{2} + 3 a + 14\right)\cdot 19^{9} +O(19^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 16 a^{3} + 5 a^{2} + 14 a + 17 + \left(a^{3} + 16 a^{2} + 11 a + 12\right)\cdot 19 + \left(16 a^{3} + 12 a^{2} + 17 a + 15\right)\cdot 19^{2} + \left(7 a^{3} + 17 a^{2} + 17 a + 6\right)\cdot 19^{3} + \left(14 a^{3} + 11 a^{2} + 15 a + 8\right)\cdot 19^{4} + \left(8 a^{3} + 10 a^{2} + 4 a + 16\right)\cdot 19^{5} + \left(13 a^{3} + 2 a^{2} + 14 a\right)\cdot 19^{6} + \left(7 a^{3} + 17 a^{2} + 16 a + 18\right)\cdot 19^{7} + \left(3 a^{3} + 15 a^{2} + a + 13\right)\cdot 19^{8} + \left(6 a^{3} + 6 a^{2} + 14 a + 4\right)\cdot 19^{9} +O(19^{10})\)
| |
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 |
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(1,8)(2,6)(3,7)(5,9)$ | $0$ |
$36$ | $2$ | $(1,7)(3,9)(4,6)$ | $-2$ |
$8$ | $3$ | $(1,3,2)(4,5,9)(6,7,8)$ | $-1$ |
$24$ | $3$ | $(2,7,9)(3,5,6)$ | $2$ |
$48$ | $3$ | $(1,9,8)(2,5,7)(3,4,6)$ | $-1$ |
$54$ | $4$ | $(1,2,8,6)(3,5,7,9)$ | $0$ |
$72$ | $6$ | $(1,4,9,7,6,3)(2,8,5)$ | $1$ |
$72$ | $6$ | $(1,2,4,7,8,9)(3,6)$ | $0$ |
$54$ | $8$ | $(1,3,8,2,9,5,7,4)$ | $0$ |
$54$ | $8$ | $(1,5,8,4,9,3,7,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.