Properties

Label 1.109.9t1.1
Dimension 1
Group $C_9$
Conductor $ 109 $
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$1$
Group:$C_9$
Conductor:$109 $
Artin number field: Splitting field of $f= x^{9} - x^{8} - 48 x^{7} + 73 x^{6} + 660 x^{5} - 1454 x^{4} - 2149 x^{3} + 8350 x^{2} - 7432 x + 2008 $ over $\Q$
Size of Galois orbit: 6
Smallest containing permutation representation: $C_9$
Parity: Even

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{3} + 4 x + 17 $
Roots:
$r_{ 1 }$ $=$ $ 15 a^{2} + 11 a + 8 + \left(2 a^{2} + 2 a + 6\right)\cdot 19 + \left(17 a^{2} + 5 a + 7\right)\cdot 19^{2} + \left(4 a^{2} + 5 a + 16\right)\cdot 19^{3} + \left(a^{2} + 9\right)\cdot 19^{4} + \left(2 a^{2} + 18 a + 10\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 17 a^{2} + 7 + \left(7 a^{2} + 17 a + 7\right)\cdot 19 + \left(5 a^{2} + 12 a + 1\right)\cdot 19^{2} + \left(6 a^{2} + a + 1\right)\cdot 19^{3} + \left(7 a^{2} + 14 a + 7\right)\cdot 19^{4} + \left(7 a^{2} + 5 a + 18\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 5 a^{2} + 15 a + 15 + \left(16 a^{2} + 2 a + 1\right)\cdot 19 + \left(8 a^{2} + 6 a + 1\right)\cdot 19^{2} + \left(13 a^{2} + a + 18\right)\cdot 19^{3} + \left(10 a^{2} + 12 a + 13\right)\cdot 19^{4} + \left(13 a^{2} + 18 a + 13\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 4 }$ $=$ $ a + 18 + \left(15 a^{2} + 13 a + 12\right)\cdot 19 + \left(9 a^{2} + 14 a + 4\right)\cdot 19^{2} + \left(9 a^{2} + 18 a + 8\right)\cdot 19^{3} + \left(9 a^{2} + 13 a + 14\right)\cdot 19^{4} + \left(12 a^{2} + 7 a + 12\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 17 a^{2} + 13 a + 9 + \left(5 a^{2} + 15 a + 18\right)\cdot 19 + \left(15 a^{2} + a + 11\right)\cdot 19^{2} + \left(2 a^{2} + 11 a + 8\right)\cdot 19^{3} + \left(8 a^{2} + 12 a + 13\right)\cdot 19^{4} + \left(8 a^{2} + 3 a + 12\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 6 a^{2} + 8 a + 3 + \left(8 a^{2} + 18 a + 2\right)\cdot 19 + \left(15 a^{2} + 9\right)\cdot 19^{2} + \left(7 a^{2} + 12 a + 11\right)\cdot 19^{3} + \left(10 a^{2} + 4 a + 2\right)\cdot 19^{4} + \left(9 a^{2} + 14 a + 5\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 16 a^{2} + 10 a + \left(15 a^{2} + 7\right)\cdot 19 + \left(13 a^{2} + 11 a + 14\right)\cdot 19^{2} + \left(2 a^{2} + 6 a + 14\right)\cdot 19^{3} + \left(13 a + 4\right)\cdot 19^{4} + \left(16 a^{2} + 15 a + 1\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 16 a^{2} + 2 a + 10 + \left(4 a^{2} + 17 a + 17\right)\cdot 19 + \left(15 a^{2} + 15 a + 12\right)\cdot 19^{2} + \left(2 a^{2} + 4 a + 15\right)\cdot 19^{3} + \left(10 a^{2} + 2 a + 9\right)\cdot 19^{4} + \left(3 a^{2} + 18 a + 1\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$
$r_{ 9 }$ $=$ $ 3 a^{2} + 16 a + 7 + \left(18 a^{2} + 7 a + 2\right)\cdot 19 + \left(12 a^{2} + 7 a + 13\right)\cdot 19^{2} + \left(6 a^{2} + 14 a\right)\cdot 19^{3} + \left(18 a^{2} + 2 a\right)\cdot 19^{4} + \left(2 a^{2} + 12 a\right)\cdot 19^{5} +O\left(19^{ 6 }\right)$

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

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

Character values on conjugacy classes

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