Basic invariants
Dimension: | $8$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(17703899136\)\(\medspace = 2^{12} \cdot 3^{6} \cdot 7^{2} \cdot 11^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 9.1.2366667072.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T213 |
Parity: | even |
Projective image: | $S_3\wr S_3$ |
Projective field: | Galois closure of 9.1.2366667072.1 |
Galois action
Roots of defining polynomial
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:
$r_{ 1 }$ | $=$ | \( 16 + 14\cdot 31 + 3\cdot 31^{2} + 29\cdot 31^{3} + 6\cdot 31^{5} + 7\cdot 31^{6} + 13\cdot 31^{7} + 3\cdot 31^{8} + 16\cdot 31^{9} +O(31^{10})\) |
$r_{ 2 }$ | $=$ | \( 17 a^{3} + 19 a^{2} + 3 a + 25 + \left(25 a^{3} + 20 a^{2} + 5 a\right)\cdot 31 + \left(28 a^{3} + 12 a^{2} + 22 a + 10\right)\cdot 31^{2} + \left(21 a^{3} + 30 a^{2} + 22\right)\cdot 31^{3} + \left(4 a^{3} + 7 a^{2} + 20 a + 24\right)\cdot 31^{4} + \left(10 a^{3} + 7 a^{2} + 23 a + 10\right)\cdot 31^{5} + \left(5 a^{3} + 24 a + 23\right)\cdot 31^{6} + \left(21 a^{3} + 25 a^{2} + 8 a + 13\right)\cdot 31^{7} + \left(23 a^{3} + 25 a^{2} + 27 a + 4\right)\cdot 31^{8} + \left(29 a^{3} + 5 a^{2} + 29 a + 12\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 3 }$ | $=$ | \( 9 a^{3} + 17 a^{2} + 18 a + 26 + \left(17 a^{3} + 17 a^{2} + 14 a + 27\right)\cdot 31 + \left(7 a^{2} + 14 a + 26\right)\cdot 31^{2} + \left(10 a^{3} + 28 a^{2} + 5 a + 30\right)\cdot 31^{3} + \left(14 a^{3} + 7 a^{2} + 20 a + 28\right)\cdot 31^{4} + \left(3 a^{3} + 20 a^{2} + 11 a\right)\cdot 31^{5} + \left(30 a^{3} + 14 a^{2} + a + 6\right)\cdot 31^{6} + \left(5 a^{3} + 6 a^{2} + 15 a + 17\right)\cdot 31^{7} + \left(a^{3} + 11 a^{2} + 18 a + 15\right)\cdot 31^{8} + \left(14 a^{3} + 8 a^{2} + 11 a + 6\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 4 }$ | $=$ | \( 24 a^{3} + 5 a^{2} + 10 a + 2 + \left(15 a^{3} + 22 a^{2} + 26 a + 1\right)\cdot 31 + \left(30 a^{3} + 12 a^{2} + 5 a + 22\right)\cdot 31^{2} + \left(18 a^{3} + 11 a^{2} + 10 a + 4\right)\cdot 31^{3} + \left(10 a^{3} + 15 a^{2} + 29 a + 11\right)\cdot 31^{4} + \left(11 a^{3} + 23 a^{2} + 18 a + 7\right)\cdot 31^{5} + \left(15 a^{3} + 11 a^{2} + 9 a + 26\right)\cdot 31^{6} + \left(4 a^{2} + a + 25\right)\cdot 31^{7} + \left(2 a^{3} + 29 a^{2} + 22 a + 5\right)\cdot 31^{8} + \left(23 a^{3} + 26 a^{2} + 14 a + 3\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 5 }$ | $=$ | \( 14 a^{3} + 12 a^{2} + 28 a + 25 + \left(5 a^{3} + 10 a^{2} + 25 a + 7\right)\cdot 31 + \left(2 a^{3} + 18 a^{2} + 8 a + 24\right)\cdot 31^{2} + \left(9 a^{3} + 30 a + 24\right)\cdot 31^{3} + \left(26 a^{3} + 23 a^{2} + 10 a + 11\right)\cdot 31^{4} + \left(20 a^{3} + 23 a^{2} + 7 a + 24\right)\cdot 31^{5} + \left(25 a^{3} + 30 a^{2} + 6 a + 18\right)\cdot 31^{6} + \left(9 a^{3} + 5 a^{2} + 22 a + 19\right)\cdot 31^{7} + \left(7 a^{3} + 5 a^{2} + 3 a + 9\right)\cdot 31^{8} + \left(a^{3} + 25 a^{2} + a + 24\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 6 }$ | $=$ | \( a^{3} + 23 a^{2} + 2 a + 12 + \left(19 a^{3} + 28 a^{2} + 28 a + 5\right)\cdot 31 + \left(26 a^{3} + 21 a^{2} + 19 a + 21\right)\cdot 31^{2} + \left(23 a^{3} + 21 a^{2} + 1\right)\cdot 31^{3} + \left(7 a^{3} + 20 a^{2} + 28 a + 12\right)\cdot 31^{4} + \left(29 a^{3} + 27 a^{2} + 23 a + 2\right)\cdot 31^{5} + \left(7 a^{2} + 9\right)\cdot 31^{6} + \left(a^{3} + 14 a^{2} + a + 30\right)\cdot 31^{7} + \left(21 a^{3} + 24 a^{2} + 25 a + 16\right)\cdot 31^{8} + \left(20 a^{3} + 22 a^{2} + 13 a\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 7 }$ | $=$ | \( 20 a^{3} + 30 a^{2} + 9 a + 7 + \left(24 a^{3} + 28 a^{2} + 27 a + 9\right)\cdot 31 + \left(23 a^{3} + 12 a^{2} + 13 a + 19\right)\cdot 31^{2} + \left(9 a^{3} + 11 a^{2} + 8 a + 2\right)\cdot 31^{3} + \left(17 a^{3} + 2 a^{2} + 28 a + 10\right)\cdot 31^{4} + \left(25 a^{3} + 13 a^{2} + 14 a + 7\right)\cdot 31^{5} + \left(6 a^{3} + 4 a^{2} + 7 a + 21\right)\cdot 31^{6} + \left(15 a^{3} + 18 a^{2} + 22 a + 21\right)\cdot 31^{7} + \left(20 a^{2} + 21 a + 5\right)\cdot 31^{8} + \left(16 a^{3} + 24 a^{2} + 8\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 8 }$ | $=$ | \( 30 a^{3} + 8 a^{2} + 29 a + 12 + \left(11 a^{3} + 2 a^{2} + 2 a + 20\right)\cdot 31 + \left(4 a^{3} + 9 a^{2} + 11 a + 29\right)\cdot 31^{2} + \left(7 a^{3} + 9 a^{2} + 30 a + 14\right)\cdot 31^{3} + \left(23 a^{3} + 10 a^{2} + 2 a + 11\right)\cdot 31^{4} + \left(a^{3} + 3 a^{2} + 7 a + 23\right)\cdot 31^{5} + \left(30 a^{3} + 23 a^{2} + 30 a + 24\right)\cdot 31^{6} + \left(29 a^{3} + 16 a^{2} + 29 a + 24\right)\cdot 31^{7} + \left(9 a^{3} + 6 a^{2} + 5 a + 27\right)\cdot 31^{8} + \left(10 a^{3} + 8 a^{2} + 17 a + 24\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 9 }$ | $=$ | \( 9 a^{3} + 10 a^{2} + 25 a + \left(4 a^{3} + 24 a^{2} + 24 a + 6\right)\cdot 31 + \left(7 a^{3} + 28 a^{2} + 27 a + 29\right)\cdot 31^{2} + \left(23 a^{3} + 10 a^{2} + 6 a + 23\right)\cdot 31^{3} + \left(19 a^{3} + 5 a^{2} + 15 a + 12\right)\cdot 31^{4} + \left(21 a^{3} + 5 a^{2} + 16 a + 10\right)\cdot 31^{5} + \left(9 a^{3} + 12 a + 18\right)\cdot 31^{6} + \left(9 a^{3} + 2 a^{2} + 23 a + 19\right)\cdot 31^{7} + \left(27 a^{3} + a^{2} + 30 a + 3\right)\cdot 31^{8} + \left(8 a^{3} + 2 a^{2} + 3 a + 28\right)\cdot 31^{9} +O(31^{10})\) |
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 values |
$c1$ | |||
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(2,4)$ | $0$ |
$18$ | $2$ | $(1,2)(4,6)(8,9)$ | $4$ |
$27$ | $2$ | $(1,6)(2,4)(3,5)$ | $0$ |
$27$ | $2$ | $(2,4)(3,5)$ | $0$ |
$54$ | $2$ | $(1,3)(2,4)(5,6)(7,8)$ | $0$ |
$6$ | $3$ | $(3,5,7)$ | $-4$ |
$8$ | $3$ | $(1,8,6)(2,9,4)(3,7,5)$ | $-1$ |
$12$ | $3$ | $(1,6,8)(3,5,7)$ | $2$ |
$72$ | $3$ | $(1,3,2)(4,6,5)(7,9,8)$ | $2$ |
$54$ | $4$ | $(2,3,4,5)(7,9)$ | $0$ |
$162$ | $4$ | $(2,3,4,5)(6,8)(7,9)$ | $0$ |
$36$ | $6$ | $(1,2)(3,5,7)(4,6)(8,9)$ | $-2$ |
$36$ | $6$ | $(2,3,9,7,4,5)$ | $-2$ |
$36$ | $6$ | $(2,4)(3,5,7)$ | $0$ |
$36$ | $6$ | $(1,6,8)(2,4)(3,5,7)$ | $0$ |
$54$ | $6$ | $(1,8,6)(2,4)(3,5)$ | $0$ |
$72$ | $6$ | $(1,4,6,9,8,2)(3,5,7)$ | $1$ |
$108$ | $6$ | $(1,3,6,5,8,7)(2,4)$ | $0$ |
$216$ | $6$ | $(1,3,4,6,5,2)(7,9,8)$ | $0$ |
$144$ | $9$ | $(1,3,9,8,7,4,6,5,2)$ | $-1$ |
$108$ | $12$ | $(1,6,8)(2,3,4,5)(7,9)$ | $0$ |