Basic invariants
Dimension: | $4$ |
Group: | $C_2 \wr S_4$ |
Conductor: | \(5080\)\(\medspace = 2^{3} \cdot 5 \cdot 127 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.5161280.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2 \wr S_4$ |
Parity: | even |
Determinant: | 1.5080.2t1.b.a |
Projective image: | $C_2^3:S_4$ |
Projective stem field: | Galois closure of 8.4.645160000.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} - x^{4} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: \( x^{2} + 63x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 62 a + 23 + \left(28 a + 7\right)\cdot 67 + 2\cdot 67^{2} + \left(19 a + 41\right)\cdot 67^{3} + \left(44 a + 55\right)\cdot 67^{4} + \left(23 a + 61\right)\cdot 67^{5} + \left(51 a + 40\right)\cdot 67^{6} + \left(50 a + 39\right)\cdot 67^{7} + \left(9 a + 41\right)\cdot 67^{8} + \left(43 a + 34\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 2 }$ | $=$ | \( 9 + 13\cdot 67 + 41\cdot 67^{3} + 8\cdot 67^{4} + 25\cdot 67^{5} + 67^{6} + 5\cdot 67^{7} + 41\cdot 67^{8} + 54\cdot 67^{9} +O(67^{10})\) |
$r_{ 3 }$ | $=$ | \( 22 a + 6 + \left(23 a + 44\right)\cdot 67 + \left(34 a + 2\right)\cdot 67^{2} + \left(25 a + 18\right)\cdot 67^{3} + \left(54 a + 34\right)\cdot 67^{4} + \left(38 a + 8\right)\cdot 67^{5} + 56\cdot 67^{6} + \left(60 a + 63\right)\cdot 67^{7} + \left(29 a + 36\right)\cdot 67^{8} + \left(54 a + 23\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 4 }$ | $=$ | \( 5 a + 3 + \left(38 a + 61\right)\cdot 67 + \left(66 a + 41\right)\cdot 67^{2} + \left(47 a + 49\right)\cdot 67^{3} + \left(22 a + 12\right)\cdot 67^{4} + \left(43 a + 45\right)\cdot 67^{5} + \left(15 a + 21\right)\cdot 67^{6} + \left(16 a + 57\right)\cdot 67^{7} + \left(57 a + 29\right)\cdot 67^{8} + \left(23 a + 63\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 5 }$ | $=$ | \( 52 + 6\cdot 67 + 5\cdot 67^{2} + 46\cdot 67^{3} + 12\cdot 67^{4} + 46\cdot 67^{5} + 36\cdot 67^{6} + 57\cdot 67^{7} + 10\cdot 67^{8} + 45\cdot 67^{9} +O(67^{10})\) |
$r_{ 6 }$ | $=$ | \( 45 a + 27 + \left(43 a + 48\right)\cdot 67 + \left(32 a + 49\right)\cdot 67^{2} + \left(41 a + 18\right)\cdot 67^{3} + \left(12 a + 25\right)\cdot 67^{4} + \left(28 a + 42\right)\cdot 67^{5} + \left(66 a + 19\right)\cdot 67^{6} + \left(6 a + 35\right)\cdot 67^{7} + \left(37 a + 29\right)\cdot 67^{8} + \left(12 a + 10\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 7 }$ | $=$ | \( 38 a + 32 + \left(65 a + 65\right)\cdot 67 + \left(40 a + 33\right)\cdot 67^{2} + \left(19 a + 41\right)\cdot 67^{3} + \left(57 a + 21\right)\cdot 67^{4} + \left(56 a + 1\right)\cdot 67^{5} + \left(36 a + 34\right)\cdot 67^{6} + \left(46 a + 30\right)\cdot 67^{7} + \left(31 a + 32\right)\cdot 67^{8} + \left(59 a + 15\right)\cdot 67^{9} +O(67^{10})\) |
$r_{ 8 }$ | $=$ | \( 29 a + 50 + \left(a + 21\right)\cdot 67 + \left(26 a + 65\right)\cdot 67^{2} + \left(47 a + 11\right)\cdot 67^{3} + \left(9 a + 30\right)\cdot 67^{4} + \left(10 a + 37\right)\cdot 67^{5} + \left(30 a + 57\right)\cdot 67^{6} + \left(20 a + 45\right)\cdot 67^{7} + \left(35 a + 45\right)\cdot 67^{8} + \left(7 a + 20\right)\cdot 67^{9} +O(67^{10})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-4$ |
$4$ | $2$ | $(3,4)$ | $2$ |
$4$ | $2$ | $(2,5)(3,4)(7,8)$ | $-2$ |
$6$ | $2$ | $(1,6)(3,4)$ | $0$ |
$12$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
$12$ | $2$ | $(1,2)(5,6)$ | $2$ |
$12$ | $2$ | $(1,7)(2,5)(3,4)(6,8)$ | $-2$ |
$24$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
$32$ | $3$ | $(1,3,7)(4,8,6)$ | $1$ |
$12$ | $4$ | $(1,3,6,4)(2,7,5,8)$ | $0$ |
$12$ | $4$ | $(1,2,6,5)$ | $2$ |
$12$ | $4$ | $(1,6)(2,5)(3,8,4,7)$ | $-2$ |
$24$ | $4$ | $(1,3,6,4)(2,7)(5,8)$ | $0$ |
$24$ | $4$ | $(1,2,6,5)(3,4)$ | $0$ |
$48$ | $4$ | $(1,2,3,7)(4,8,6,5)$ | $0$ |
$32$ | $6$ | $(2,7,3,5,8,4)$ | $1$ |
$32$ | $6$ | $(1,3,7)(2,5)(4,8,6)$ | $-1$ |
$32$ | $6$ | $(1,3,8,6,4,7)(2,5)$ | $-1$ |
$48$ | $8$ | $(1,7,3,5,6,8,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.