Basic invariants
Dimension: | $8$ |
Group: | $\PGL(2,7)$ |
Conductor: | \(52115390874114183\)\(\medspace = 3^{6} \cdot 7^{9} \cdot 11^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.1063579405594167.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 21T20 |
Parity: | odd |
Determinant: | 1.7.2t1.a.a |
Projective image: | $\PGL(2,7)$ |
Projective stem field: | Galois closure of 8.2.1063579405594167.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 77x^{5} - 154x^{4} - 231x^{3} + 385x^{2} - 1067x - 1947 \) . |
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:
$r_{ 1 }$ | $=$ | \( 8 a + \left(2 a^{2} + a + 25\right)\cdot 29 + \left(16 a^{2} + 11 a + 12\right)\cdot 29^{2} + \left(15 a^{2} + 22 a + 14\right)\cdot 29^{3} + \left(6 a^{2} + 20\right)\cdot 29^{4} + \left(12 a^{2} + 27 a + 27\right)\cdot 29^{5} + \left(13 a^{2} + 3 a + 1\right)\cdot 29^{6} + \left(3 a^{2} + 13 a + 28\right)\cdot 29^{7} + \left(2 a^{2} + 18 a + 15\right)\cdot 29^{8} + \left(19 a^{2} + 4\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 2 }$ | $=$ | \( 22 a^{2} + 10 + \left(6 a^{2} + 19 a + 2\right)\cdot 29 + \left(13 a^{2} + 25 a + 9\right)\cdot 29^{2} + \left(12 a^{2} + 10\right)\cdot 29^{3} + \left(13 a^{2} + 24 a + 10\right)\cdot 29^{4} + \left(20 a^{2} + 15 a + 19\right)\cdot 29^{5} + \left(7 a^{2} + 22 a + 13\right)\cdot 29^{6} + \left(4 a^{2} + 19 a + 19\right)\cdot 29^{7} + \left(22 a^{2} + 7 a + 13\right)\cdot 29^{8} + \left(15 a^{2} + 21 a + 19\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 3 }$ | $=$ | \( 7 a^{2} + 21 a + 19 + \left(20 a^{2} + 8 a + 10\right)\cdot 29 + \left(28 a^{2} + 21 a + 10\right)\cdot 29^{2} + \left(5 a + 14\right)\cdot 29^{3} + \left(9 a^{2} + 4 a + 4\right)\cdot 29^{4} + \left(25 a^{2} + 15 a + 16\right)\cdot 29^{5} + \left(7 a^{2} + 2 a + 23\right)\cdot 29^{6} + \left(21 a^{2} + 25 a + 22\right)\cdot 29^{7} + \left(4 a^{2} + 2 a + 9\right)\cdot 29^{8} + \left(23 a^{2} + 7 a\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 4 }$ | $=$ | \( 2 a^{2} + 23 a + 6 + \left(3 a^{2} + 23 a + 23\right)\cdot 29 + \left(17 a^{2} + 10\right)\cdot 29^{2} + \left(6 a^{2} + 15 a + 21\right)\cdot 29^{3} + \left(a^{2} + 9 a + 12\right)\cdot 29^{4} + \left(7 a^{2} + 5\right)\cdot 29^{5} + \left(21 a^{2} + 16\right)\cdot 29^{6} + \left(14 a^{2} + 23 a + 13\right)\cdot 29^{7} + \left(25 a^{2} + 6 a + 26\right)\cdot 29^{8} + \left(11 a^{2} + a\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 5 }$ | $=$ | \( 25 a^{2} + 23 a + 27 + \left(6 a^{2} + 22 a + 8\right)\cdot 29 + \left(23 a^{2} + 20 a + 9\right)\cdot 29^{2} + \left(8 a^{2} + 6 a + 24\right)\cdot 29^{3} + \left(5 a^{2} + 4 a + 27\right)\cdot 29^{4} + \left(10 a^{2} + 27 a + 28\right)\cdot 29^{5} + \left(12 a^{2} + 16 a + 13\right)\cdot 29^{6} + \left(3 a^{2} + 21 a + 27\right)\cdot 29^{7} + \left(17 a^{2} + 17 a + 24\right)\cdot 29^{8} + \left(23 a^{2} + 13 a + 6\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 6 }$ | $=$ | \( 27 + 28\cdot 29 + 25\cdot 29^{2} + 12\cdot 29^{3} + 18\cdot 29^{4} + 21\cdot 29^{5} + 27\cdot 29^{6} + 12\cdot 29^{7} + 9\cdot 29^{8} + 6\cdot 29^{9} +O(29^{10})\) |
$r_{ 7 }$ | $=$ | \( 2 a^{2} + 12 a + 6 + \left(19 a^{2} + 11 a + 25\right)\cdot 29 + \left(17 a^{2} + 7 a + 1\right)\cdot 29^{2} + \left(13 a^{2} + 7 a + 21\right)\cdot 29^{3} + \left(22 a^{2} + 15 a + 21\right)\cdot 29^{4} + \left(11 a^{2} + a + 11\right)\cdot 29^{5} + \left(24 a^{2} + 12 a + 20\right)\cdot 29^{6} + \left(10 a^{2} + 13 a + 27\right)\cdot 29^{7} + \left(15 a^{2} + 4 a + 12\right)\cdot 29^{8} + \left(22 a^{2} + 14 a + 5\right)\cdot 29^{9} +O(29^{10})\) |
$r_{ 8 }$ | $=$ | \( 23 + 20\cdot 29 + 6\cdot 29^{2} + 26\cdot 29^{3} + 28\cdot 29^{4} + 13\cdot 29^{5} + 27\cdot 29^{6} + 21\cdot 29^{7} + 2\cdot 29^{8} + 14\cdot 29^{9} +O(29^{10})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$21$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
$28$ | $2$ | $(1,6)(2,4)(5,7)$ | $2$ |
$56$ | $3$ | $(1,5,4)(2,6,7)$ | $-1$ |
$42$ | $4$ | $(1,2,5,4)(3,8,7,6)$ | $0$ |
$56$ | $6$ | $(1,2,5,6,4,7)$ | $-1$ |
$48$ | $7$ | $(1,3,2,5,6,8,4)$ | $1$ |
$42$ | $8$ | $(1,3,2,8,5,7,4,6)$ | $0$ |
$42$ | $8$ | $(1,8,4,3,5,6,2,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.