Basic invariants
Dimension: | $16$ |
Group: | $((C_3^2:Q_8):C_3):C_2$ |
Conductor: | \(101\!\cdots\!625\)\(\medspace = 3^{34} \cdot 5^{14} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.3.18160335421875.5 |
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 9.3.18160335421875.5 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 9x^{7} - 15x^{6} + 63x^{4} + 120x^{3} + 108x^{2} + 45x + 10 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{4} + 3x^{2} + 16x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 6 a^{3} + 10 a^{2} + 26 a + 4 + \left(13 a^{3} + 13 a^{2} + 9\right)\cdot 31 + \left(10 a^{3} + 7 a^{2} + 2 a + 28\right)\cdot 31^{2} + \left(21 a^{3} + 21 a^{2} + 10\right)\cdot 31^{3} + \left(13 a^{3} + 16 a^{2} + 19 a + 29\right)\cdot 31^{4} + \left(29 a^{3} + 19 a^{2} + 4 a + 15\right)\cdot 31^{5} + \left(20 a^{3} + 16 a^{2} + 13 a + 19\right)\cdot 31^{6} + \left(19 a^{3} + 26 a^{2} + 2 a + 4\right)\cdot 31^{7} + \left(14 a^{3} + 21 a + 2\right)\cdot 31^{8} + \left(18 a^{3} + 9 a^{2} + 9 a + 6\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 2 }$ | $=$ | \( 18 a^{3} + a^{2} + 22 a + 23 + \left(5 a^{3} + 8 a^{2} + 18 a + 30\right)\cdot 31 + \left(14 a^{3} + 25 a^{2} + 15 a + 16\right)\cdot 31^{2} + \left(18 a^{3} + 23 a^{2} + 4 a + 8\right)\cdot 31^{3} + \left(3 a^{3} + 12 a^{2} + 16 a + 10\right)\cdot 31^{4} + \left(13 a^{3} + 14 a^{2} + a + 29\right)\cdot 31^{5} + \left(26 a^{3} + 20 a^{2} + 17 a + 7\right)\cdot 31^{6} + \left(16 a^{3} + 19 a^{2} + 11 a + 28\right)\cdot 31^{7} + \left(3 a^{3} + a^{2} + 17 a + 26\right)\cdot 31^{8} + \left(18 a^{3} + 20 a + 27\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 3 }$ | $=$ | \( 25 + 31 + 18\cdot 31^{2} + 25\cdot 31^{3} + 10\cdot 31^{4} + 19\cdot 31^{5} + 30\cdot 31^{6} + 7\cdot 31^{7} + 27\cdot 31^{8} +O(31^{10})\) |
$r_{ 4 }$ | $=$ | \( 2 a^{3} + 26 a^{2} + 19 a + 11 + \left(14 a^{3} + 5 a^{2} + 17 a + 8\right)\cdot 31 + \left(11 a^{3} + 7 a^{2} + 23 a + 9\right)\cdot 31^{2} + \left(24 a^{3} + 11 a^{2} + 13 a + 1\right)\cdot 31^{3} + \left(6 a^{3} + 17 a^{2} + 2 a + 25\right)\cdot 31^{4} + \left(22 a + 8\right)\cdot 31^{5} + \left(21 a^{3} + 11 a^{2} + 5 a + 27\right)\cdot 31^{6} + \left(9 a^{3} + 26 a^{2} + 24 a + 23\right)\cdot 31^{7} + \left(21 a^{3} + 16 a^{2} + a + 28\right)\cdot 31^{8} + \left(5 a^{3} + 16 a + 25\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 5 }$ | $=$ | \( 2 a^{3} + 19 a^{2} + 21 a + 16 + \left(5 a^{3} + 17 a^{2} + 17 a + 26\right)\cdot 31 + \left(14 a^{3} + 19 a^{2} + 4 a + 13\right)\cdot 31^{2} + \left(22 a^{3} + 24 a^{2} + 24 a + 29\right)\cdot 31^{3} + \left(13 a^{3} + 8 a^{2} + 8 a + 17\right)\cdot 31^{4} + \left(21 a^{3} + 22 a^{2} + 24 a + 1\right)\cdot 31^{5} + \left(6 a^{3} + 20 a^{2} + 2 a + 25\right)\cdot 31^{6} + \left(17 a^{3} + 15 a^{2} + 23 a + 20\right)\cdot 31^{7} + \left(25 a^{3} + 18 a^{2} + 20 a + 4\right)\cdot 31^{8} + \left(22 a^{3} + 2 a^{2} + 30 a + 2\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 6 }$ | $=$ | \( 24 a^{3} + 28 a^{2} + 23 a + 27 + \left(2 a^{3} + 28 a^{2} + 18 a + 12\right)\cdot 31 + \left(15 a^{3} + 21 a^{2} + 12 a + 7\right)\cdot 31^{2} + \left(20 a^{3} + 4 a^{2} + 24 a + 4\right)\cdot 31^{3} + \left(11 a^{3} + 9 a^{2} + 3 a + 24\right)\cdot 31^{4} + \left(25 a^{3} + 9 a^{2} + 18 a + 13\right)\cdot 31^{5} + \left(11 a^{3} + a^{2} + a + 5\right)\cdot 31^{6} + \left(29 a^{3} + 20 a^{2} + 16 a + 24\right)\cdot 31^{7} + \left(11 a^{3} + 19 a^{2} + 28 a + 30\right)\cdot 31^{8} + \left(15 a^{3} + 10 a^{2} + 15 a + 10\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 7 }$ | $=$ | \( 21 a^{3} + 7 a^{2} + 27 a + 9 + \left(29 a^{3} + 25 a^{2} + 25 a + 23\right)\cdot 31 + \left(25 a^{3} + 27 a^{2} + 12\right)\cdot 31^{2} + \left(24 a^{3} + 4 a^{2} + 24 a + 28\right)\cdot 31^{3} + \left(27 a^{3} + 19 a^{2}\right)\cdot 31^{4} + \left(10 a^{3} + 19 a^{2} + 11 a + 26\right)\cdot 31^{5} + \left(13 a^{3} + 13 a^{2} + 9 a + 16\right)\cdot 31^{6} + \left(15 a^{3} + 24 a^{2} + 12 a + 12\right)\cdot 31^{7} + \left(25 a^{2} + 18 a + 9\right)\cdot 31^{8} + \left(15 a^{3} + 18 a^{2} + 5 a + 10\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 8 }$ | $=$ | \( 14 a^{3} + 22 a^{2} + 11 a + 22 + \left(30 a^{3} + 8 a^{2} + 12 a + 4\right)\cdot 31 + \left(29 a^{3} + 19 a^{2} + 24 a + 27\right)\cdot 31^{2} + \left(26 a^{3} + 15 a^{2} + 21 a + 20\right)\cdot 31^{3} + \left(30 a^{3} + 16 a + 24\right)\cdot 31^{4} + \left(29 a^{3} + 28 a^{2} + 7 a + 19\right)\cdot 31^{5} + \left(18 a^{3} + a^{2} + 11 a + 14\right)\cdot 31^{6} + \left(21 a^{3} + 11 a^{2} + 19 a + 10\right)\cdot 31^{7} + \left(21 a^{3} + 3 a^{2} + 20 a + 30\right)\cdot 31^{8} + \left(16 a^{3} + a^{2} + 23 a + 27\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 9 }$ | $=$ | \( 6 a^{3} + 11 a^{2} + 6 a + 18 + \left(23 a^{3} + 16 a^{2} + 12 a + 6\right)\cdot 31 + \left(2 a^{3} + 26 a^{2} + 9 a + 21\right)\cdot 31^{2} + \left(27 a^{3} + 17 a^{2} + 11 a + 25\right)\cdot 31^{3} + \left(15 a^{3} + 8 a^{2} + 25 a + 11\right)\cdot 31^{4} + \left(24 a^{3} + 10 a^{2} + 3 a + 20\right)\cdot 31^{5} + \left(4 a^{3} + 7 a^{2} + a + 7\right)\cdot 31^{6} + \left(25 a^{3} + 11 a^{2} + 15 a + 22\right)\cdot 31^{7} + \left(24 a^{3} + 6 a^{2} + 26 a + 25\right)\cdot 31^{8} + \left(11 a^{3} + 19 a^{2} + a + 11\right)\cdot 31^{9} +O(31^{10})\) |
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 |
$1$ | $1$ | $()$ | $16$ |
$9$ | $2$ | $(1,5)(2,8)(4,7)(6,9)$ | $0$ |
$36$ | $2$ | $(3,5)(6,8)(7,9)$ | $0$ |
$8$ | $3$ | $(1,4,2)(3,6,9)(5,8,7)$ | $-2$ |
$24$ | $3$ | $(1,9,8)(3,4,7)$ | $-2$ |
$48$ | $3$ | $(1,3,2)(4,5,8)(6,9,7)$ | $1$ |
$54$ | $4$ | $(1,4,5,7)(2,9,8,6)$ | $0$ |
$72$ | $6$ | $(1,4,2)(3,8,9,5,6,7)$ | $0$ |
$72$ | $6$ | $(1,3,8,7,9,4)(2,5)$ | $0$ |
$54$ | $8$ | $(1,9,4,8,5,6,7,2)$ | $0$ |
$54$ | $8$ | $(1,6,4,2,5,9,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.