Properties

Label 1.189.9t1.a
Dimension $1$
Group $C_9$
Conductor $189$
Indicator $0$

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Basic invariants

Dimension:$1$
Group:$C_9$
Conductor:\(189\)\(\medspace = 3^{3} \cdot 7 \)
Artin number field: Galois closure of 9.9.3691950281939241.1
Galois orbit size: $6$
Smallest permutation container: $C_9$
Parity: even
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{3} + x + 14\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 4 a^{2} + 6 a + 14 + \left(9 a^{2} + 11\right)\cdot 17 + \left(6 a^{2} + 5 a + 15\right)\cdot 17^{2} + \left(14 a^{2} + 5 a + 3\right)\cdot 17^{3} + \left(12 a^{2} + 3 a + 14\right)\cdot 17^{4} + \left(7 a^{2} + 13 a + 10\right)\cdot 17^{5} + \left(a^{2} + 7 a + 6\right)\cdot 17^{6} + \left(5 a^{2} + 13 a + 3\right)\cdot 17^{7} + \left(7 a^{2} + 5 a + 16\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 13 a^{2} + 4 a + 3 + \left(9 a^{2} + 15 a + 12\right)\cdot 17 + \left(8 a^{2} + 4 a + 5\right)\cdot 17^{2} + \left(6 a^{2} + 12 a + 4\right)\cdot 17^{3} + \left(2 a^{2} + 4 a + 7\right)\cdot 17^{4} + \left(16 a^{2} + 16 a + 16\right)\cdot 17^{5} + \left(16 a^{2} + 2 a + 16\right)\cdot 17^{6} + \left(8 a^{2} + 12 a + 5\right)\cdot 17^{7} + \left(10 a^{2} + 9 a + 1\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 3 }$ $=$ \( a^{2} + 14 a + 12 + \left(11 a^{2} + 10 a + 1\right)\cdot 17 + \left(15 a^{2} + 16 a + 16\right)\cdot 17^{2} + \left(14 a^{2} + 13 a + 9\right)\cdot 17^{3} + \left(6 a^{2} + 5 a + 4\right)\cdot 17^{4} + \left(16 a^{2} + 8 a + 5\right)\cdot 17^{5} + \left(12 a^{2} + 8 a + 14\right)\cdot 17^{6} + \left(2 a^{2} + 3 a + 1\right)\cdot 17^{7} + \left(10 a^{2} + 1\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 5 a^{2} + 8 a + 9 + \left(4 a^{2} + 9 a + 8\right)\cdot 17 + \left(14 a^{2} + 9 a + 9\right)\cdot 17^{2} + \left(2 a^{2} + 10 a + 7\right)\cdot 17^{3} + \left(11 a^{2} + 5 a + 7\right)\cdot 17^{4} + \left(5 a^{2} + 9\right)\cdot 17^{5} + \left(12 a^{2} + a + 2\right)\cdot 17^{6} + \left(14 a^{2} + 5 a + 4\right)\cdot 17^{7} + \left(15 a^{2} + 2 a + 16\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 16 a^{2} + 5 a + 5 + \left(2 a^{2} + 9 a + 13\right)\cdot 17 + \left(11 a^{2} + 2 a + 1\right)\cdot 17^{2} + \left(7 a^{2} + 11 a + 5\right)\cdot 17^{3} + \left(3 a^{2} + 6 a + 2\right)\cdot 17^{4} + \left(12 a^{2} + 8\right)\cdot 17^{5} + \left(4 a^{2} + 13 a + 14\right)\cdot 17^{6} + \left(10 a^{2} + 16 a + 6\right)\cdot 17^{7} + \left(7 a^{2} + 4 a + 16\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 8 a + \left(5 a^{2} + 2 a + 9\right)\cdot 17 + \left(3 a^{2} + 6 a + 13\right)\cdot 17^{2} + \left(13 a^{2} + 11 a + 8\right)\cdot 17^{3} + \left(16 a^{2} + 8 a + 5\right)\cdot 17^{4} + \left(16 a^{2} + 8 a + 11\right)\cdot 17^{5} + \left(10 a^{2} + 10 a + 1\right)\cdot 17^{6} + \left(13 a^{2} + 9 a + 9\right)\cdot 17^{7} + \left(7 a^{2} + 4 a + 16\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 2 a^{2} + 5 a + 7 + \left(9 a^{2} + 3 a\right)\cdot 17 + \left(14 a^{2} + 15 a + 4\right)\cdot 17^{2} + \left(4 a^{2} + 10 a + 3\right)\cdot 17^{3} + \left(3 a^{2} + 11 a + 2\right)\cdot 17^{4} + \left(16 a^{2} + 12 a + 5\right)\cdot 17^{5} + \left(4 a^{2} + 5 a + 3\right)\cdot 17^{6} + \left(6 a^{2} + 11 a + 4\right)\cdot 17^{7} + \left(a^{2} + 16 a + 12\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 12 a^{2} + 14 a + 8 + \left(13 a^{2} + 5 a + 3\right)\cdot 17 + \left(11 a^{2} + 12 a + 2\right)\cdot 17^{2} + \left(4 a^{2} + 14 a + 3\right)\cdot 17^{3} + \left(14 a^{2} + 7 a + 15\right)\cdot 17^{4} + \left(9 a^{2} + 12 a\right)\cdot 17^{5} + \left(2 a^{2} + 13\right)\cdot 17^{6} + \left(9 a^{2} + 11\right)\cdot 17^{7} + \left(16 a^{2} + 11 a + 16\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 15 a^{2} + 4 a + 10 + \left(2 a^{2} + 11 a + 7\right)\cdot 17 + \left(16 a^{2} + 12 a + 16\right)\cdot 17^{2} + \left(15 a^{2} + 11 a + 4\right)\cdot 17^{3} + \left(13 a^{2} + 13 a + 9\right)\cdot 17^{4} + 12 a\cdot 17^{5} + \left(a^{2} + 12\right)\cdot 17^{6} + \left(14 a^{2} + 13 a + 3\right)\cdot 17^{7} + \left(7 a^{2} + 12 a + 5\right)\cdot 17^{8} +O(17^{9})\)  Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(1,3,8)(2,4,5)(6,9,7)$
$(1,5,9,3,2,7,8,4,6)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character values
$c1$ $c2$ $c3$ $c4$ $c5$ $c6$
$1$ $1$ $()$ $1$ $1$ $1$ $1$ $1$ $1$
$1$ $3$ $(1,3,8)(2,4,5)(6,9,7)$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$
$1$ $3$ $(1,8,3)(2,5,4)(6,7,9)$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$ $-\zeta_{9}^{3} - 1$ $\zeta_{9}^{3}$
$1$ $9$ $(1,5,9,3,2,7,8,4,6)$ $\zeta_{9}$ $\zeta_{9}^{2}$ $\zeta_{9}^{4}$ $\zeta_{9}^{5}$ $-\zeta_{9}^{4} - \zeta_{9}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$
$1$ $9$ $(1,9,2,8,6,5,3,7,4)$ $\zeta_{9}^{2}$ $\zeta_{9}^{4}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $\zeta_{9}$ $\zeta_{9}^{5}$ $-\zeta_{9}^{4} - \zeta_{9}$
$1$ $9$ $(1,2,6,3,4,9,8,5,7)$ $\zeta_{9}^{4}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $-\zeta_{9}^{4} - \zeta_{9}$ $\zeta_{9}^{2}$ $\zeta_{9}$ $\zeta_{9}^{5}$
$1$ $9$ $(1,7,5,8,9,4,3,6,2)$ $\zeta_{9}^{5}$ $\zeta_{9}$ $\zeta_{9}^{2}$ $-\zeta_{9}^{4} - \zeta_{9}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $\zeta_{9}^{4}$
$1$ $9$ $(1,4,7,3,5,6,8,2,9)$ $-\zeta_{9}^{4} - \zeta_{9}$ $\zeta_{9}^{5}$ $\zeta_{9}$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $\zeta_{9}^{4}$ $\zeta_{9}^{2}$
$1$ $9$ $(1,6,4,8,7,2,3,9,5)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$ $-\zeta_{9}^{4} - \zeta_{9}$ $\zeta_{9}^{5}$ $\zeta_{9}^{4}$ $\zeta_{9}^{2}$ $\zeta_{9}$
The blue line marks the conjugacy class containing complex conjugation.