Basic invariants
Dimension: | $9$ |
Group: | $\PSL(2,8)$ |
Conductor: | \(514147280633856\)\(\medspace = 2^{14} \cdot 3^{22} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.1.514147280633856.1 |
Galois orbit size: | $3$ |
Smallest permutation container: | $\PSL(2,8)$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $\SL(2,8)$ |
Projective stem field: | Galois closure of 9.1.514147280633856.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 12x^{6} - 18x^{5} + 36x^{2} - 27x - 128 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{3} + x + 14 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 a^{2} + 10 a + 8 + \left(9 a^{2} + 7 a + 1\right)\cdot 17 + \left(10 a^{2} + 11 a + 1\right)\cdot 17^{2} + \left(2 a^{2} + 8 a + 9\right)\cdot 17^{3} + \left(15 a^{2} + 8\right)\cdot 17^{4} + \left(2 a^{2} + 2 a + 1\right)\cdot 17^{5} + \left(4 a^{2} + 12\right)\cdot 17^{6} + \left(10 a^{2} + 10 a + 12\right)\cdot 17^{7} + \left(2 a^{2} + 13 a\right)\cdot 17^{8} + \left(6 a^{2} + 12 a + 6\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 2 }$ | $=$ | \( 16 a^{2} + 8 a + 8 + \left(15 a^{2} + 4 a + 8\right)\cdot 17 + \left(4 a^{2} + 5\right)\cdot 17^{2} + \left(11 a^{2} + 16 a + 15\right)\cdot 17^{3} + \left(12 a^{2} + 16\right)\cdot 17^{4} + \left(6 a^{2} + a + 6\right)\cdot 17^{5} + \left(9 a^{2} + 11 a + 1\right)\cdot 17^{6} + \left(13 a^{2} + 14 a + 15\right)\cdot 17^{7} + \left(10 a^{2} + 2 a + 14\right)\cdot 17^{8} + \left(12 a^{2} + 9 a + 4\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 3 }$ | $=$ | \( 4 a^{2} + 13 a + 10 + \left(5 a^{2} + 13 a + 4\right)\cdot 17 + \left(15 a^{2} + a + 8\right)\cdot 17^{2} + \left(a^{2} + 8 a + 14\right)\cdot 17^{3} + \left(10 a^{2} + 8 a + 16\right)\cdot 17^{4} + \left(16 a^{2} + 7 a + 8\right)\cdot 17^{5} + \left(11 a^{2} + 6 a + 3\right)\cdot 17^{6} + \left(12 a^{2} + 4 a + 2\right)\cdot 17^{7} + \left(4 a^{2} + 3 a + 2\right)\cdot 17^{8} + \left(10 a^{2} + 14\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 4 }$ | $=$ | \( 8 a^{2} + 14 a + 14 + \left(10 a^{2} + 15 a + 4\right)\cdot 17 + \left(11 a^{2} + 7 a + 4\right)\cdot 17^{2} + \left(5 a^{2} + 9 a\right)\cdot 17^{3} + \left(16 a^{2} + 11 a + 8\right)\cdot 17^{4} + \left(13 a^{2} + 15 a\right)\cdot 17^{5} + \left(7 a^{2} + 15 a + 6\right)\cdot 17^{6} + \left(8 a^{2} + 13 a\right)\cdot 17^{7} + \left(14 a^{2} + 13 a + 6\right)\cdot 17^{8} + \left(8 a^{2} + 6 a + 2\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 5 }$ | $=$ | \( a^{2} + 12 a + 8 + \left(16 a^{2} + 3 a\right)\cdot 17 + \left(4 a^{2} + 12 a + 7\right)\cdot 17^{2} + \left(9 a^{2} + 6 a + 2\right)\cdot 17^{3} + \left(2 a^{2} + 6\right)\cdot 17^{4} + \left(10 a^{2} + 14 a + 10\right)\cdot 17^{5} + \left(12 a + 1\right)\cdot 17^{6} + \left(11 a^{2} + 3 a + 1\right)\cdot 17^{7} + \left(9 a^{2} + 15 a + 11\right)\cdot 17^{8} + 15 a^{2} 17^{9} +O(17^{10})\) |
$r_{ 6 }$ | $=$ | \( 12 a^{2} + 5 a + 9 + \left(7 a^{2} + a\right)\cdot 17 + \left(7 a^{2} + 5 a + 16\right)\cdot 17^{2} + \left(11 a^{2} + 8 a + 14\right)\cdot 17^{3} + \left(12 a^{2} + 5 a + 6\right)\cdot 17^{4} + \left(16 a^{2} + 9 a + 16\right)\cdot 17^{5} + \left(3 a^{2} + 12 a + 11\right)\cdot 17^{6} + \left(12 a^{2} + 11 a + 2\right)\cdot 17^{7} + \left(16 a^{2} + 7 a + 10\right)\cdot 17^{8} + \left(5 a^{2} + 8 a + 11\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 7 }$ | $=$ | \( 3 a^{2} + 2 a + 3 + \left(8 a + 1\right)\cdot 17 + \left(16 a^{2} + 16\right)\cdot 17^{2} + \left(2 a^{2} + 14\right)\cdot 17^{3} + \left(6 a^{2} + 11 a + 13\right)\cdot 17^{4} + \left(14 a^{2} + 5 a + 14\right)\cdot 17^{5} + \left(8 a^{2} + 4 a + 3\right)\cdot 17^{6} + \left(11 a^{2} + 12 a + 2\right)\cdot 17^{7} + \left(14 a^{2} + 12 a + 3\right)\cdot 17^{8} + \left(4 a^{2} + 12 a + 5\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 8 }$ | $=$ | \( 10 a^{2} + 12 a + 4 + \left(7 a^{2} + 13 a + 14\right)\cdot 17 + \left(8 a + 13\right)\cdot 17^{2} + \left(8 a + 7\right)\cdot 17^{3} + \left(5 a^{2} + 4 a\right)\cdot 17^{4} + 13 a^{2} 17^{5} + \left(16 a^{2} + 7 a + 12\right)\cdot 17^{6} + \left(11 a^{2} + 5 a + 2\right)\cdot 17^{7} + \left(8 a^{2} + 2\right)\cdot 17^{8} + \left(12 a^{2} + a + 16\right)\cdot 17^{9} +O(17^{10})\) |
$r_{ 9 }$ | $=$ | \( 12 a^{2} + 9 a + 4 + \left(12 a^{2} + 16 a + 15\right)\cdot 17 + \left(13 a^{2} + 2 a + 12\right)\cdot 17^{2} + \left(5 a^{2} + 2 a + 5\right)\cdot 17^{3} + \left(4 a^{2} + 8 a + 7\right)\cdot 17^{4} + \left(7 a^{2} + 12 a + 8\right)\cdot 17^{5} + \left(4 a^{2} + 14 a + 15\right)\cdot 17^{6} + \left(10 a^{2} + 8 a + 11\right)\cdot 17^{7} + \left(2 a^{2} + 15 a\right)\cdot 17^{8} + \left(8 a^{2} + 15 a + 7\right)\cdot 17^{9} +O(17^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
$1$ | $1$ | $()$ | $9$ |
$63$ | $2$ | $(1,8)(2,3)(5,6)(7,9)$ | $1$ |
$56$ | $3$ | $(1,5,2)(3,7,6)(4,8,9)$ | $0$ |
$72$ | $7$ | $(1,9,2,6,5,8,3)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$72$ | $7$ | $(1,2,5,3,9,6,8)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$72$ | $7$ | $(1,6,3,2,8,9,5)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$56$ | $9$ | $(1,3,6,9,8,5,4,2,7)$ | $0$ |
$56$ | $9$ | $(1,6,8,4,7,3,9,5,2)$ | $0$ |
$56$ | $9$ | $(1,8,7,9,2,6,4,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.