Basic invariants
Dimension: | $6$ |
Group: | $\PGL(2,7)$ |
Conductor: | \(295740809071407\)\(\medspace = 3^{6} \cdot 7^{5} \cdot 17^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.295740809071407.1 |
Galois orbit size: | $2$ |
Smallest permutation container: | 14T16 |
Parity: | odd |
Determinant: | 1.7.2t1.a.a |
Projective image: | $\PGL(2,7)$ |
Projective stem field: | Galois closure of 8.2.295740809071407.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 3x^{7} + 5x^{6} - 8x^{5} - 60x^{4} - 16x^{3} + 496x^{2} + 384x - 256 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{3} + 2x + 27 \)
Roots:
$r_{ 1 }$ | $=$ | \( 3 + 22\cdot 29 + 7\cdot 29^{2} + 26\cdot 29^{3} + 3\cdot 29^{4} + 2\cdot 29^{5} + 12\cdot 29^{6} + 6\cdot 29^{7} + 12\cdot 29^{8} + 5\cdot 29^{9} +O(29^{10})\) |
$r_{ 2 }$ | $=$ | \( 21 a^{2} + 10 a + 23 + \left(26 a^{2} + 28 a + 20\right)\cdot 29 + \left(13 a^{2} + 4 a + 17\right)\cdot 29^{2} + \left(14 a^{2} + 17 a + 17\right)\cdot 29^{3} + \left(22 a^{2} + 15 a + 1\right)\cdot 29^{4} + \left(20 a^{2} + 25 a + 13\right)\cdot 29^{5} + \left(12 a^{2} + 21 a + 25\right)\cdot 29^{6} + \left(a^{2} + a + 14\right)\cdot 29^{7} + \left(7 a^{2} + 3 a + 3\right)\cdot 29^{8} + \left(14 a^{2} + 16 a + 16\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 3 }$ | $=$ | \( 16 a^{2} + 24 a + 26 + \left(15 a^{2} + 27 a + 5\right)\cdot 29 + \left(25 a^{2} + 27 a + 4\right)\cdot 29^{2} + \left(21 a^{2} + 13 a + 8\right)\cdot 29^{3} + \left(23 a^{2} + 20 a + 3\right)\cdot 29^{4} + \left(25 a^{2} + 20 a + 10\right)\cdot 29^{5} + \left(19 a^{2} + 9 a + 25\right)\cdot 29^{6} + \left(8 a^{2} + 6 a + 14\right)\cdot 29^{7} + \left(28 a^{2} + 26 a + 12\right)\cdot 29^{8} + \left(3 a^{2} + 25 a + 2\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 4 }$ | $=$ | \( 21 + 29 + 29^{2} + 16\cdot 29^{3} + 12\cdot 29^{4} + 3\cdot 29^{5} + 17\cdot 29^{6} + 8\cdot 29^{7} + 18\cdot 29^{8} + 15\cdot 29^{9} +O(29^{10})\) |
$r_{ 5 }$ | $=$ | \( 17 a^{2} + 10 a + 11 + \left(13 a^{2} + 16 a + 15\right)\cdot 29 + \left(25 a^{2} + 27 a + 12\right)\cdot 29^{2} + \left(17 a^{2} + 7 a + 11\right)\cdot 29^{3} + \left(17 a^{2} + a + 17\right)\cdot 29^{4} + \left(21 a^{2} + a + 12\right)\cdot 29^{5} + \left(7 a^{2} + 2 a + 11\right)\cdot 29^{6} + \left(15 a^{2} + 2 a + 2\right)\cdot 29^{7} + \left(24 a^{2} + 24 a + 9\right)\cdot 29^{8} + \left(a^{2} + 14 a + 27\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 6 }$ | $=$ | \( 21 a^{2} + 24 a + 23 + \left(15 a^{2} + a + 15\right)\cdot 29 + \left(18 a^{2} + 25 a + 4\right)\cdot 29^{2} + \left(21 a^{2} + 26 a + 27\right)\cdot 29^{3} + \left(11 a^{2} + 21 a + 25\right)\cdot 29^{4} + \left(11 a^{2} + 11 a + 19\right)\cdot 29^{5} + \left(25 a^{2} + 26 a + 3\right)\cdot 29^{6} + \left(18 a^{2} + 20 a + 9\right)\cdot 29^{7} + \left(22 a^{2} + 28 a + 24\right)\cdot 29^{8} + \left(10 a^{2} + 15 a + 1\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 7 }$ | $=$ | \( 16 a^{2} + 12 a + \left(21 a^{2} + 26\right)\cdot 29 + \left(16 a^{2} + 2 a\right)\cdot 29^{2} + \left(28 a^{2} + 17 a + 16\right)\cdot 29^{3} + \left(2 a^{2} + 18 a + 7\right)\cdot 29^{4} + \left(18 a^{2} + 19 a + 27\right)\cdot 29^{5} + \left(10 a^{2} + 17 a + 24\right)\cdot 29^{6} + \left(4 a^{2} + 2 a + 16\right)\cdot 29^{7} + \left(15 a^{2} + 28 a + 25\right)\cdot 29^{8} + \left(27 a^{2} + 21 a + 22\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 8 }$ | $=$ | \( 25 a^{2} + 7 a + 12 + \left(22 a^{2} + 12 a + 8\right)\cdot 29 + \left(15 a^{2} + 28 a + 9\right)\cdot 29^{2} + \left(11 a^{2} + 3 a + 22\right)\cdot 29^{3} + \left(8 a^{2} + 9 a + 14\right)\cdot 29^{4} + \left(18 a^{2} + 8 a + 27\right)\cdot 29^{5} + \left(10 a^{2} + 9 a + 24\right)\cdot 29^{6} + \left(9 a^{2} + 24 a + 13\right)\cdot 29^{7} + \left(18 a^{2} + 5 a + 10\right)\cdot 29^{8} + \left(28 a^{2} + 21 a + 24\right)\cdot 29^{9} +O(29^{10})\) |
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 |
$1$ | $1$ | $()$ | $6$ |
$21$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $2$ |
$28$ | $2$ | $(1,3)(2,7)(4,5)$ | $0$ |
$56$ | $3$ | $(1,5,7)(2,3,4)$ | $0$ |
$42$ | $4$ | $(1,5,7,4)(2,6,8,3)$ | $0$ |
$56$ | $6$ | $(1,2,5,3,7,4)$ | $0$ |
$48$ | $7$ | $(1,2,8,5,7,6,4)$ | $-1$ |
$42$ | $8$ | $(1,3,5,2,7,6,4,8)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
$42$ | $8$ | $(1,2,4,3,7,8,5,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.