Basic invariants
Dimension: | $8$ |
Group: | $C_3^2:C_8$ |
Conductor: | \(173946175488\)\(\medspace = 2^{31} \cdot 3^{4} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.173946175488.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_3^2:C_8$ |
Parity: | even |
Determinant: | 1.8.2t1.a.a |
Projective image: | $F_9$ |
Projective stem field: | Galois closure of 9.1.173946175488.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 4x^{8} + 8x^{7} - 32x^{5} + 80x^{4} - 104x^{3} + 80x^{2} - 34x + 8 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{4} + 3x^{2} + 19x + 5 \)
Roots:
$r_{ 1 }$ | $=$ | \( 6 a^{3} + 4 a^{2} + 15 a + 12 + \left(17 a^{3} + 8 a^{2} + 19 a + 21\right)\cdot 23 + \left(9 a^{3} + 7 a^{2} + 4 a + 15\right)\cdot 23^{2} + \left(3 a^{3} + 13 a^{2} + 21 a + 20\right)\cdot 23^{3} + \left(a^{3} + 9 a^{2} + 17 a + 15\right)\cdot 23^{4} + \left(14 a^{3} + 4 a^{2} + 6 a + 9\right)\cdot 23^{5} + \left(7 a^{3} + 7 a^{2} + 7\right)\cdot 23^{6} + \left(9 a^{3} + 12 a^{2} + 22 a + 22\right)\cdot 23^{7} + \left(13 a^{3} + 5 a^{2} + 12 a + 17\right)\cdot 23^{8} + \left(18 a^{3} + 7 a^{2} + 11 a + 6\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 2 }$ | $=$ | \( 6 a^{3} + 3 a^{2} + 6 + \left(18 a^{3} + 7 a^{2} + 15 a + 9\right)\cdot 23 + \left(20 a^{3} + 20 a^{2} + 11 a + 20\right)\cdot 23^{2} + \left(22 a^{3} + 11 a^{2} + 15 a + 15\right)\cdot 23^{3} + \left(19 a^{3} + 21 a^{2} + 4\right)\cdot 23^{4} + \left(15 a^{3} + 5 a^{2} + 4 a + 7\right)\cdot 23^{5} + \left(8 a^{3} + 9 a^{2} + 4 a\right)\cdot 23^{6} + \left(13 a^{3} + 12 a^{2} + 20 a + 2\right)\cdot 23^{7} + \left(13 a^{3} + 11 a^{2} + 17 a + 19\right)\cdot 23^{8} + \left(19 a^{3} + 2 a^{2} + 13 a + 10\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 3 }$ | $=$ | \( 22 a^{3} + 6 a^{2} + 18 a + 13 + \left(15 a^{3} + 2 a^{2} + 22 a + 5\right)\cdot 23 + \left(3 a^{3} + 21 a^{2} + 4 a + 19\right)\cdot 23^{2} + \left(19 a^{3} + 19 a^{2} + a + 1\right)\cdot 23^{3} + \left(8 a^{3} + 19 a^{2} + 12 a + 20\right)\cdot 23^{4} + \left(6 a^{3} + 16 a^{2} + 5 a + 16\right)\cdot 23^{5} + \left(14 a^{3} + 5 a^{2} + 11 a + 19\right)\cdot 23^{6} + \left(13 a^{3} + 19 a^{2} + 13 a + 1\right)\cdot 23^{7} + \left(4 a^{3} + 8 a^{2} + 18\right)\cdot 23^{8} + \left(22 a^{3} + 14 a^{2} + 11 a + 5\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 4 }$ | $=$ | \( 19 a^{3} + 7 a^{2} + 14 a + 12 + \left(7 a^{3} + 19 a^{2} + 13 a + 1\right)\cdot 23 + \left(11 a^{3} + 7 a^{2} + 18 a + 22\right)\cdot 23^{2} + \left(18 a^{3} + 12 a^{2} + 3\right)\cdot 23^{3} + \left(12 a^{3} + 16 a^{2} + 14\right)\cdot 23^{4} + \left(a^{3} + 19 a^{2} + 18 a + 15\right)\cdot 23^{5} + \left(18 a^{3} + 14 a^{2} + 15 a + 12\right)\cdot 23^{6} + \left(10 a^{3} + 2 a^{2} + 5 a + 5\right)\cdot 23^{7} + \left(13 a^{3} + 21 a^{2} + 5 a + 19\right)\cdot 23^{8} + \left(20 a^{3} + a^{2} + 16 a + 15\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 5 }$ | $=$ | \( 22 a^{3} + 6 a^{2} + 22 a + 13 + \left(4 a^{3} + 16 a^{2} + 12 a + 13\right)\cdot 23 + \left(21 a^{3} + 9 a^{2} + 17 a + 4\right)\cdot 23^{2} + \left(4 a^{3} + 22 a + 11\right)\cdot 23^{3} + \left(15 a + 5\right)\cdot 23^{4} + \left(a^{3} + 5 a^{2} + 15 a + 20\right)\cdot 23^{5} + \left(6 a^{3} + 18 a^{2} + 18 a + 18\right)\cdot 23^{6} + \left(12 a^{3} + 11 a^{2} + 4 a + 5\right)\cdot 23^{7} + \left(14 a^{3} + 10 a^{2} + 4 a + 1\right)\cdot 23^{8} + \left(7 a^{3} + 22 a^{2} + 7 a\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 6 }$ | $=$ | \( 19 a^{3} + 20 a^{2} + 8 a + 4 + \left(14 a^{3} + 12 a^{2} + 18 a + 9\right)\cdot 23 + \left(13 a^{3} + 16 a^{2} + 3 a + 16\right)\cdot 23^{2} + \left(22 a^{3} + 22 a^{2} + 4\right)\cdot 23^{3} + \left(9 a^{3} + 22 a^{2} + 11 a + 2\right)\cdot 23^{4} + \left(18 a^{3} + 14 a^{2} + 3 a + 17\right)\cdot 23^{5} + \left(5 a^{3} + 3 a^{2} + 19\right)\cdot 23^{6} + \left(11 a^{3} + 17 a^{2} + 17 a + 1\right)\cdot 23^{7} + \left(12 a^{3} + 9 a^{2} + 8 a + 18\right)\cdot 23^{8} + \left(2 a^{3} + 15 a^{2} + 20 a + 5\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 7 }$ | $=$ | \( 15 a^{3} + 16 a^{2} + 19 a + 10 + \left(6 a^{3} + 16 a + 12\right)\cdot 23 + \left(17 a^{3} + 11 a^{2} + 5 a + 2\right)\cdot 23^{2} + \left(15 a^{3} + 9 a + 6\right)\cdot 23^{3} + \left(17 a^{3} + 5 a^{2} + 13 a + 10\right)\cdot 23^{4} + \left(9 a^{3} + 8 a^{2} + 21 a + 21\right)\cdot 23^{5} + \left(10 a^{3} + 12 a^{2} + 16 a\right)\cdot 23^{6} + \left(14 a^{3} + 7 a^{2} + 5 a + 10\right)\cdot 23^{7} + \left(20 a^{3} + 2 a^{2} + 18 a + 19\right)\cdot 23^{8} + \left(18 a^{3} + 21 a^{2} + 15 a + 5\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 8 }$ | $=$ | \( 6 a^{3} + 7 a^{2} + 19 a + 12 + \left(6 a^{3} + 2 a^{2} + 18 a + 3\right)\cdot 23 + \left(17 a^{3} + 21 a^{2} + a\right)\cdot 23^{2} + \left(7 a^{3} + 10 a^{2} + 21 a + 17\right)\cdot 23^{3} + \left(21 a^{3} + 19 a^{2} + 20 a + 20\right)\cdot 23^{4} + \left(a^{3} + 16 a^{2} + 16 a + 8\right)\cdot 23^{5} + \left(21 a^{3} + 20 a^{2} + a + 4\right)\cdot 23^{6} + \left(6 a^{3} + 8 a^{2} + 3 a + 8\right)\cdot 23^{7} + \left(22 a^{3} + 22 a^{2} + a + 4\right)\cdot 23^{8} + \left(4 a^{3} + 6 a^{2} + 19 a + 16\right)\cdot 23^{9} +O(23^{10})\) |
$r_{ 9 }$ | $=$ | \( 14 + 15\cdot 23 + 13\cdot 23^{2} + 10\cdot 23^{3} + 21\cdot 23^{4} + 20\cdot 23^{5} + 7\cdot 23^{6} + 11\cdot 23^{7} + 20\cdot 23^{8} + 23^{9} +O(23^{10})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(2,6)(3,9)(4,7)(5,8)$ | $0$ |
$8$ | $3$ | $(1,5,8)(2,9,7)(3,6,4)$ | $-1$ |
$9$ | $4$ | $(2,3,6,9)(4,8,7,5)$ | $0$ |
$9$ | $4$ | $(2,9,6,3)(4,5,7,8)$ | $0$ |
$9$ | $8$ | $(2,7,3,5,6,4,9,8)$ | $0$ |
$9$ | $8$ | $(2,5,9,7,6,8,3,4)$ | $0$ |
$9$ | $8$ | $(2,4,3,8,6,7,9,5)$ | $0$ |
$9$ | $8$ | $(2,8,9,4,6,5,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.