Basic invariants
Dimension: | $4$ |
Group: | $C_2 \wr S_4$ |
Conductor: | \(4303\)\(\medspace = 13 \cdot 331 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.1424293.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2 \wr S_4$ |
Parity: | odd |
Determinant: | 1.4303.2t1.a.a |
Projective image: | $C_2^3:S_4$ |
Projective stem field: | Galois closure of 8.4.3129171721.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 2x^{6} - x^{5} + x^{4} - x^{3} + 2x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: \( x^{2} + 58x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 40 + 15\cdot 59 + 35\cdot 59^{2} + 53\cdot 59^{3} + 15\cdot 59^{4} + 54\cdot 59^{5} + 34\cdot 59^{6} + 54\cdot 59^{7} + 5\cdot 59^{8} + 44\cdot 59^{9} +O(59^{10})\) |
$r_{ 2 }$ | $=$ | \( 7 a + 11 + \left(14 a + 4\right)\cdot 59 + \left(42 a + 25\right)\cdot 59^{2} + \left(40 a + 58\right)\cdot 59^{3} + \left(47 a + 8\right)\cdot 59^{4} + \left(19 a + 50\right)\cdot 59^{5} + \left(9 a + 54\right)\cdot 59^{6} + \left(4 a + 12\right)\cdot 59^{7} + \left(20 a + 28\right)\cdot 59^{8} + \left(40 a + 29\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 3 }$ | $=$ | \( 52 a + 50 + \left(55 a + 29\right)\cdot 59 + \left(47 a + 48\right)\cdot 59^{2} + \left(32 a + 42\right)\cdot 59^{3} + \left(33 a + 25\right)\cdot 59^{4} + \left(28 a + 3\right)\cdot 59^{5} + \left(32 a + 16\right)\cdot 59^{6} + \left(44 a + 21\right)\cdot 59^{7} + \left(45 a + 36\right)\cdot 59^{8} + \left(32 a + 25\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 4 }$ | $=$ | \( 48 a + 28 + \left(17 a + 7\right)\cdot 59 + \left(41 a + 47\right)\cdot 59^{2} + \left(20 a + 36\right)\cdot 59^{3} + \left(51 a + 32\right)\cdot 59^{4} + \left(8 a + 19\right)\cdot 59^{5} + \left(35 a + 4\right)\cdot 59^{6} + \left(25 a + 23\right)\cdot 59^{7} + \left(12 a + 29\right)\cdot 59^{8} + \left(28 a + 44\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 5 }$ | $=$ | \( 7 a + 43 + \left(3 a + 33\right)\cdot 59 + \left(11 a + 40\right)\cdot 59^{2} + \left(26 a + 27\right)\cdot 59^{3} + \left(25 a + 26\right)\cdot 59^{4} + \left(30 a + 57\right)\cdot 59^{5} + \left(26 a + 19\right)\cdot 59^{6} + \left(14 a + 33\right)\cdot 59^{7} + \left(13 a + 37\right)\cdot 59^{8} + \left(26 a + 12\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 6 }$ | $=$ | \( 11 a + 17 + \left(41 a + 36\right)\cdot 59 + \left(17 a + 11\right)\cdot 59^{2} + \left(38 a + 16\right)\cdot 59^{3} + \left(7 a + 4\right)\cdot 59^{4} + \left(50 a + 36\right)\cdot 59^{5} + \left(23 a + 30\right)\cdot 59^{6} + \left(33 a + 13\right)\cdot 59^{7} + \left(46 a + 16\right)\cdot 59^{8} + \left(30 a + 1\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 7 }$ | $=$ | \( 52 a + 18 + \left(44 a + 11\right)\cdot 59 + \left(16 a + 53\right)\cdot 59^{2} + \left(18 a + 56\right)\cdot 59^{3} + \left(11 a + 15\right)\cdot 59^{4} + \left(39 a + 22\right)\cdot 59^{5} + \left(49 a + 44\right)\cdot 59^{6} + \left(54 a + 7\right)\cdot 59^{7} + \left(38 a + 44\right)\cdot 59^{8} + \left(18 a + 49\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 8 }$ | $=$ | \( 31 + 38\cdot 59 + 33\cdot 59^{2} + 2\cdot 59^{3} + 47\cdot 59^{4} + 51\cdot 59^{5} + 30\cdot 59^{6} + 10\cdot 59^{7} + 38\cdot 59^{8} + 28\cdot 59^{9} +O(59^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
$1$ | $1$ | $()$ | $4$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
$4$ | $2$ | $(1,8)$ | $2$ |
$4$ | $2$ | $(1,8)(2,7)(3,6)$ | $-2$ |
$6$ | $2$ | $(1,8)(4,5)$ | $0$ |
$12$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
$12$ | $2$ | $(2,4)(5,7)$ | $2$ |
$12$ | $2$ | $(1,8)(2,7)(3,4)(5,6)$ | $-2$ |
$24$ | $2$ | $(1,8)(2,4)(5,7)$ | $0$ |
$32$ | $3$ | $(1,3,4)(5,8,6)$ | $1$ |
$12$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
$12$ | $4$ | $(2,5,7,4)$ | $2$ |
$12$ | $4$ | $(1,6,8,3)(2,7)(4,5)$ | $-2$ |
$24$ | $4$ | $(1,5,8,4)(2,3)(6,7)$ | $0$ |
$24$ | $4$ | $(1,8)(2,5,7,4)$ | $0$ |
$48$ | $4$ | $(1,3,4,2)(5,7,8,6)$ | $0$ |
$32$ | $6$ | $(1,7,6,8,2,3)$ | $1$ |
$32$ | $6$ | $(1,3,4)(2,7)(5,8,6)$ | $-1$ |
$32$ | $6$ | $(1,6,5,8,3,4)(2,7)$ | $-1$ |
$48$ | $8$ | $(1,7,5,6,8,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.