Basic invariants
Dimension: | $3$ |
Group: | $(C_3^2:C_3):C_2$ |
Conductor: | \(2484300\)\(\medspace = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 13^{2} \) |
Artin stem field: | Galois closure of 9.3.90724673403000000.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | $(C_3^2:C_3):C_2$ |
Parity: | odd |
Determinant: | 1.3.2t1.a.a |
Projective image: | $C_3:S_3$ |
Projective field: | Galois closure of Degree 9 field |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 2x^{8} - 9x^{7} - 84x^{6} + 549x^{5} - 820x^{4} + 3991x^{3} - 23254x^{2} + 45892x - 28168 \) . |
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:
$r_{ 1 }$ | $=$ | \( 4 a^{2} + 2 a + 15 + \left(a^{2} + 9 a + 11\right)\cdot 19 + \left(13 a^{2} + 17 a + 13\right)\cdot 19^{2} + \left(14 a^{2} + 11 a + 9\right)\cdot 19^{3} + \left(5 a^{2} + 11 a + 14\right)\cdot 19^{4} + \left(14 a^{2} + 3 a + 6\right)\cdot 19^{5} + \left(9 a^{2} + 13 a + 2\right)\cdot 19^{6} + \left(2 a^{2} + 3 a + 15\right)\cdot 19^{7} + \left(18 a^{2} + 11 a + 12\right)\cdot 19^{8} + \left(9 a^{2} + 9 a + 2\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 2 }$ | $=$ | \( 14 a^{2} + 10 a + 10 + \left(17 a^{2} + 2 a + 11\right)\cdot 19 + \left(12 a^{2} + 16 a\right)\cdot 19^{2} + \left(7 a^{2} + 10 a + 10\right)\cdot 19^{3} + \left(11 a^{2} + 10\right)\cdot 19^{4} + \left(15 a^{2} + 6 a + 16\right)\cdot 19^{5} + \left(18 a^{2} + 8 a + 13\right)\cdot 19^{6} + \left(7 a^{2} + 18 a + 10\right)\cdot 19^{7} + \left(7 a^{2} + 11 a + 9\right)\cdot 19^{8} + \left(4 a^{2} + 9 a\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 3 }$ | $=$ | \( 4 a^{2} + 6 a + 11 + \left(11 a^{2} + 17 a + 18\right)\cdot 19 + \left(15 a^{2} + 15 a + 2\right)\cdot 19^{2} + \left(10 a^{2} + 17 a + 6\right)\cdot 19^{3} + \left(17 a^{2} + 9 a + 1\right)\cdot 19^{4} + \left(a^{2} + 7 a + 9\right)\cdot 19^{5} + \left(13 a^{2} + 17 a + 9\right)\cdot 19^{6} + \left(9 a^{2} + 3 a + 4\right)\cdot 19^{7} + \left(10 a^{2} + 18 a + 2\right)\cdot 19^{8} + \left(12 a^{2} + 2 a + 14\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 4 }$ | $=$ | \( 8 a^{2} + 15 a + 9 + \left(16 a^{2} + 16 a + 13\right)\cdot 19 + \left(4 a^{2} + 16 a + 18\right)\cdot 19^{2} + \left(11 a^{2} + 4 a\right)\cdot 19^{3} + \left(2 a^{2} + 8 a + 12\right)\cdot 19^{4} + \left(14 a^{2} + 7 a + 3\right)\cdot 19^{5} + \left(5 a^{2} + 12 a + 15\right)\cdot 19^{6} + \left(9 a^{2} + 9\right)\cdot 19^{7} + \left(18 a^{2} + 2 a + 4\right)\cdot 19^{8} + \left(9 a^{2} + 9 a + 7\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 5 }$ | $=$ | \( a^{2} + 7 a + 7 + \left(7 a + 2\right)\cdot 19 + \left(12 a^{2} + 4 a + 17\right)\cdot 19^{2} + \left(15 a^{2} + 15 a + 5\right)\cdot 19^{3} + \left(a^{2} + 6 a + 10\right)\cdot 19^{4} + \left(8 a^{2} + 9 a + 15\right)\cdot 19^{5} + \left(9 a^{2} + 16 a + 7\right)\cdot 19^{6} + \left(8 a^{2} + 15 a + 18\right)\cdot 19^{7} + \left(12 a^{2} + 14 a + 3\right)\cdot 19^{8} + \left(4 a^{2} + 18 a + 1\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 6 }$ | $=$ | \( 7 a^{2} + 17 a + \left(10 a^{2} + 3 a + 10\right)\cdot 19 + \left(17 a^{2} + 5 a + 14\right)\cdot 19^{2} + \left(15 a^{2} + 15 a\right)\cdot 19^{3} + \left(17 a^{2} + 2\right)\cdot 19^{4} + \left(2 a^{2} + 4 a + 18\right)\cdot 19^{5} + \left(8 a + 12\right)\cdot 19^{6} + \left(14 a + 16\right)\cdot 19^{7} + \left(9 a^{2} + 17 a + 10\right)\cdot 19^{8} + \left(15 a^{2} + 6 a + 15\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 7 }$ | $=$ | \( 18 + 6\cdot 19 + 15\cdot 19^{3} + 5\cdot 19^{4} + 4\cdot 19^{5} + 8\cdot 19^{6} + 2\cdot 19^{7} + 13\cdot 19^{8} + 8\cdot 19^{9} +O(19^{10})\) |
$r_{ 8 }$ | $=$ | \( 3 + 8\cdot 19 + 15\cdot 19^{2} + 19^{3} + 19^{4} + 15\cdot 19^{5} + 3\cdot 19^{6} + 18\cdot 19^{7} + 17\cdot 19^{8} +O(19^{10})\) |
$r_{ 9 }$ | $=$ | \( 5 + 12\cdot 19 + 11\cdot 19^{2} + 6\cdot 19^{3} + 18\cdot 19^{4} + 5\cdot 19^{5} + 2\cdot 19^{6} + 18\cdot 19^{7} + 6\cdot 19^{9} +O(19^{10})\) |
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 |
$1$ | $1$ | $()$ | $3$ |
$9$ | $2$ | $(1,6)(2,3)(4,5)$ | $1$ |
$1$ | $3$ | $(1,5,2)(3,6,4)(7,8,9)$ | $3 \zeta_{3}$ |
$1$ | $3$ | $(1,2,5)(3,4,6)(7,9,8)$ | $-3 \zeta_{3} - 3$ |
$6$ | $3$ | $(1,7,3)(2,9,4)(5,8,6)$ | $0$ |
$6$ | $3$ | $(1,8,3)(2,7,4)(5,9,6)$ | $0$ |
$6$ | $3$ | $(1,9,3)(2,8,4)(5,7,6)$ | $0$ |
$6$ | $3$ | $(3,4,6)(7,8,9)$ | $0$ |
$9$ | $6$ | $(1,4,2,6,5,3)(7,8,9)$ | $\zeta_{3}$ |
$9$ | $6$ | $(1,3,5,6,2,4)(7,9,8)$ | $-\zeta_{3} - 1$ |
The blue line marks the conjugacy class containing complex conjugation.