Basic invariants
Dimension: | $1$ |
Group: | $C_9$ |
Conductor: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
Artin field: | Galois closure of 9.9.9025761726072081.2 |
Galois orbit size: | $6$ |
Smallest permutation container: | $C_9$ |
Parity: | even |
Dirichlet character: | \(\chi_{171}(139,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{9} - 57x^{7} - 38x^{6} + 855x^{5} + 1254x^{4} - 3192x^{3} - 7524x^{2} - 4275x - 703 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a^{2} + 10 a + 7 + \left(3 a^{2} + 6 a + 8\right)\cdot 11 + \left(4 a^{2} + 3 a + 5\right)\cdot 11^{2} + \left(8 a^{2} + 6 a + 7\right)\cdot 11^{3} + \left(2 a^{2} + 6 a + 3\right)\cdot 11^{4} + \left(7 a^{2} + a + 2\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 2 }$ | $=$ | \( 6 a^{2} + 5 a + 8 + \left(3 a^{2} + 10 a + 4\right)\cdot 11 + \left(7 a^{2} + 6 a + 2\right)\cdot 11^{2} + \left(4 a^{2} + 6 a + 6\right)\cdot 11^{3} + \left(2 a^{2} + 2 a + 10\right)\cdot 11^{4} + \left(5 a^{2} + 9 a + 6\right)\cdot 11^{5} + \left(7 a^{2} + 2\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 3 }$ | $=$ | \( 10 a^{2} + 8 a + 6 + \left(7 a^{2} + 3 a + 10\right)\cdot 11 + \left(8 a^{2} + 7 a + 7\right)\cdot 11^{2} + \left(2 a^{2} + 3 a + 3\right)\cdot 11^{3} + \left(2 a^{2} + 10\right)\cdot 11^{4} + \left(8 a^{2} + 5 a + 10\right)\cdot 11^{5} + \left(6 a^{2} + 2 a + 8\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 4 }$ | $=$ | \( 2 a^{2} + 3 a + 10 + \left(a^{2} + 3 a + 8\right)\cdot 11 + \left(9 a^{2} + 3 a + 4\right)\cdot 11^{2} + \left(10 a^{2} + 3 a + 3\right)\cdot 11^{3} + \left(10 a^{2} + 6 a + 7\right)\cdot 11^{4} + \left(6 a^{2} + 4 a + 5\right)\cdot 11^{5} + \left(7 a^{2} + 3 a + 6\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 5 }$ | $=$ | \( 9 a^{2} + 4 a + 1 + \left(7 a^{2} + 5 a + 3\right)\cdot 11 + \left(a^{2} + 5 a + 2\right)\cdot 11^{2} + \left(8 a^{2} + 4 a + 7\right)\cdot 11^{3} + \left(7 a^{2} + 6\right)\cdot 11^{4} + \left(3 a^{2} + 8\right)\cdot 11^{5} + \left(10 a^{2} + 3 a + 2\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 6 }$ | $=$ | \( 10 a^{2} + 6 + \left(a^{2} + 4 a + 2\right)\cdot 11 + \left(4 a^{2} + 9\right)\cdot 11^{2} + \left(8 a^{2} + 4 a + 3\right)\cdot 11^{3} + \left(8 a^{2} + 4 a + 4\right)\cdot 11^{4} + \left(6 a^{2} + a + 5\right)\cdot 11^{5} + \left(7 a^{2} + 5 a + 6\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 7 }$ | $=$ | \( 5 a^{2} + 8 a + 3 + \left(10 a^{2} + 9 a + 10\right)\cdot 11 + \left(4 a^{2} + a + 2\right)\cdot 11^{2} + \left(5 a^{2} + 7\right)\cdot 11^{3} + 4 a\cdot 11^{4} + \left(7 a^{2} + 10 a + 2\right)\cdot 11^{5} + \left(4 a^{2} + 6 a + 6\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 8 }$ | $=$ | \( 4 a^{2} + 9 a + 9 + \left(a^{2} + 7 a + 1\right)\cdot 11 + \left(8 a^{2} + 7 a + 7\right)\cdot 11^{2} + \left(2 a^{2} + 7 a + 3\right)\cdot 11^{3} + \left(2 a^{2} + 7 a + 10\right)\cdot 11^{4} + \left(6 a^{2} + 2 a\right)\cdot 11^{5} + \left(4 a^{2} + 4 a + 6\right)\cdot 11^{6} +O(11^{7})\) |
$r_{ 9 }$ | $=$ | \( a^{2} + 8 a + 5 + \left(6 a^{2} + 3 a + 4\right)\cdot 11 + \left(6 a^{2} + 7 a + 1\right)\cdot 11^{2} + \left(3 a^{2} + 7 a + 1\right)\cdot 11^{3} + \left(6 a^{2} + 1\right)\cdot 11^{4} + \left(10 a^{2} + 10 a + 3\right)\cdot 11^{5} + \left(9 a^{2} + 5 a + 2\right)\cdot 11^{6} +O(11^{7})\) |
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 |
$1$ | $1$ | $()$ | $1$ |
$1$ | $3$ | $(1,7,5)(2,8,9)(3,6,4)$ | $\zeta_{9}^{3}$ |
$1$ | $3$ | $(1,5,7)(2,9,8)(3,4,6)$ | $-\zeta_{9}^{3} - 1$ |
$1$ | $9$ | $(1,9,3,7,2,6,5,8,4)$ | $\zeta_{9}$ |
$1$ | $9$ | $(1,3,2,5,4,9,7,6,8)$ | $\zeta_{9}^{2}$ |
$1$ | $9$ | $(1,2,4,7,8,3,5,9,6)$ | $\zeta_{9}^{4}$ |
$1$ | $9$ | $(1,6,9,5,3,8,7,4,2)$ | $\zeta_{9}^{5}$ |
$1$ | $9$ | $(1,8,6,7,9,4,5,2,3)$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
$1$ | $9$ | $(1,4,8,5,6,2,7,3,9)$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.