Basic invariants
Dimension: | $7$ |
Group: | $C_2^3:(C_7: C_3)$ |
Conductor: | \(132676581952\)\(\medspace = 2^{6} \cdot 73^{5} \) |
Artin stem field: | Galois closure of 8.0.1817487424.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | 24T283 |
Parity: | even |
Determinant: | 1.73.3t1.a.b |
Projective image: | $F_8:C_3$ |
Projective stem field: | Galois closure of 8.0.1817487424.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} + 6x^{6} - 2x^{5} + 12x^{4} + 4x^{3} + 14x^{2} + 2 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{3} + 4x + 17 \)
Roots:
$r_{ 1 }$ | $=$ | \( 11 a^{2} + 8 a + 9 + \left(13 a^{2} + 7 a + 17\right)\cdot 19 + \left(a^{2} + 9 a + 16\right)\cdot 19^{2} + \left(8 a^{2} + 11 a + 10\right)\cdot 19^{3} + \left(16 a^{2} + 13 a + 4\right)\cdot 19^{4} + \left(12 a^{2} + 8 a + 17\right)\cdot 19^{5} + \left(5 a^{2} + 17 a + 16\right)\cdot 19^{6} + \left(10 a^{2} + a + 13\right)\cdot 19^{7} + \left(4 a^{2} + 13 a + 14\right)\cdot 19^{8} + \left(13 a^{2} + 18 a + 17\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 2 }$ | $=$ | \( 8 a^{2} + a + 1 + \left(12 a^{2} + 16 a + 8\right)\cdot 19 + \left(18 a^{2} + 9 a + 11\right)\cdot 19^{2} + \left(7 a^{2} + 12 a + 10\right)\cdot 19^{3} + \left(4 a^{2} + 7 a + 10\right)\cdot 19^{4} + \left(a^{2} + 9 a + 11\right)\cdot 19^{5} + \left(7 a^{2} + 6 a + 1\right)\cdot 19^{6} + \left(15 a^{2} + 14 a + 2\right)\cdot 19^{7} + \left(14 a^{2} + 9 a + 4\right)\cdot 19^{8} + \left(12 a^{2} + 11 a + 10\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 3 }$ | $=$ | \( 8 + 2\cdot 19 + 5\cdot 19^{2} + 4\cdot 19^{3} + 17\cdot 19^{4} + 13\cdot 19^{5} + 5\cdot 19^{6} + 12\cdot 19^{7} + 10\cdot 19^{8} + 5\cdot 19^{9} +O(19^{10})\) |
$r_{ 4 }$ | $=$ | \( 13 + 12\cdot 19 + 7\cdot 19^{2} + 3\cdot 19^{3} + 8\cdot 19^{4} + 15\cdot 19^{5} + 13\cdot 19^{6} + 14\cdot 19^{7} + 3\cdot 19^{8} + 16\cdot 19^{9} +O(19^{10})\) |
$r_{ 5 }$ | $=$ | \( 12 a^{2} + 6 a + 14 + \left(13 a^{2} + 18 a + 5\right)\cdot 19 + \left(15 a^{2} + 2 a + 6\right)\cdot 19^{2} + \left(10 a^{2} + 5\right)\cdot 19^{3} + \left(13 a^{2} + 16\right)\cdot 19^{4} + \left(6 a^{2} + 2 a + 18\right)\cdot 19^{5} + \left(3 a^{2} + 16 a + 6\right)\cdot 19^{6} + \left(10 a^{2} + 9 a + 6\right)\cdot 19^{7} + \left(5 a^{2} + 10 a + 7\right)\cdot 19^{8} + \left(10 a^{2} + 3 a + 12\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 6 }$ | $=$ | \( 4 a^{2} + 11 a + 18 + \left(8 a^{2} + 17 a + 3\right)\cdot 19 + \left(14 a^{2} + 14 a + 9\right)\cdot 19^{2} + \left(4 a^{2} + 10 a + 14\right)\cdot 19^{3} + \left(17 a^{2} + 10 a\right)\cdot 19^{4} + \left(9 a^{2} + 11 a + 2\right)\cdot 19^{5} + \left(14 a^{2} + 4 a + 5\right)\cdot 19^{6} + \left(7 a^{2} + 4 a + 6\right)\cdot 19^{7} + \left(8 a^{2} + 13 a + 2\right)\cdot 19^{8} + \left(17 a^{2} + 17 a + 6\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 7 }$ | $=$ | \( 10 a + 5 + \left(12 a^{2} + 14 a + 13\right)\cdot 19 + \left(17 a^{2} + 18 a + 8\right)\cdot 19^{2} + \left(2 a^{2} + 13 a + 3\right)\cdot 19^{3} + \left(17 a^{2} + 16 a\right)\cdot 19^{4} + \left(4 a^{2} + 15\right)\cdot 19^{5} + \left(6 a^{2} + 14 a + 5\right)\cdot 19^{6} + \left(12 a^{2} + 2 a\right)\cdot 19^{7} + \left(18 a^{2} + 15 a + 8\right)\cdot 19^{8} + \left(11 a^{2} + 7 a + 14\right)\cdot 19^{9} +O(19^{10})\) |
$r_{ 8 }$ | $=$ | \( 3 a^{2} + 2 a + 9 + \left(16 a^{2} + 2 a + 12\right)\cdot 19 + \left(7 a^{2} + a + 10\right)\cdot 19^{2} + \left(3 a^{2} + 8 a + 4\right)\cdot 19^{3} + \left(7 a^{2} + 8 a + 18\right)\cdot 19^{4} + \left(2 a^{2} + 5 a\right)\cdot 19^{5} + \left(a^{2} + 17 a + 1\right)\cdot 19^{6} + \left(a^{2} + 4 a + 1\right)\cdot 19^{7} + \left(5 a^{2} + 14 a + 6\right)\cdot 19^{8} + \left(10 a^{2} + 16 a + 12\right)\cdot 19^{9} +O(19^{10})\) |
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 |
$1$ | $1$ | $()$ | $7$ |
$7$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-1$ |
$28$ | $3$ | $(3,8,6)(4,5,7)$ | $-\zeta_{3} - 1$ |
$28$ | $3$ | $(3,6,8)(4,7,5)$ | $\zeta_{3}$ |
$28$ | $6$ | $(1,2)(3,5,8,7,6,4)$ | $-\zeta_{3}$ |
$28$ | $6$ | $(1,2)(3,4,6,7,8,5)$ | $\zeta_{3} + 1$ |
$24$ | $7$ | $(1,3,4,2,5,6,7)$ | $0$ |
$24$ | $7$ | $(1,2,7,4,6,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.