# Properties

 Label 1.171.9t1.b.a Dimension $1$ Group $C_9$ Conductor $171$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_9$ Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Artin field: 9.9.9025761726072081.1 Galois orbit size: $6$ Smallest permutation container: $C_9$ Parity: even Dirichlet character: $$\chi_{171}(142,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 57 x^{7} - 38 x^{6} + 855 x^{5} + 228 x^{4} - 4902 x^{3} + 1710 x^{2} + 9063 x - 7201$$  .

The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$: $$x^{3} + 3 x + 81$$

Roots:
 $r_{ 1 }$ $=$ $$43 a^{2} + 49 a + 3 + \left(25 a^{2} + 74 a + 51\right)\cdot 83 + \left(29 a^{2} + 3 a + 58\right)\cdot 83^{2} + \left(8 a^{2} + 22 a + 16\right)\cdot 83^{3} + 37 a\cdot 83^{4} +O(83^{5})$$ $r_{ 2 }$ $=$ $$56 a^{2} + 80 a + 29 + \left(55 a^{2} + 28 a + 28\right)\cdot 83 + \left(62 a^{2} + 81 a + 42\right)\cdot 83^{2} + \left(71 a^{2} + 30 a + 60\right)\cdot 83^{3} + \left(3 a^{2} + 64 a + 7\right)\cdot 83^{4} +O(83^{5})$$ $r_{ 3 }$ $=$ $$71 a^{2} + 52 a + 59 + \left(36 a^{2} + 18 a + 73\right)\cdot 83 + \left(10 a^{2} + 77 a + 20\right)\cdot 83^{2} + \left(56 a^{2} + 81 a + 29\right)\cdot 83^{3} + \left(49 a^{2} + 52 a + 16\right)\cdot 83^{4} +O(83^{5})$$ $r_{ 4 }$ $=$ $$10 a^{2} + 37 a + 20 + \left(23 a^{2} + 24 a + 46\right)\cdot 83 + \left(4 a^{2} + 22 a + 8\right)\cdot 83^{2} + \left(80 a^{2} + 50 a + 77\right)\cdot 83^{3} + \left(8 a^{2} + 34 a + 17\right)\cdot 83^{4} +O(83^{5})$$ $r_{ 5 }$ $=$ $$17 a^{2} + 49 a + 34 + \left(4 a^{2} + 29 a + 8\right)\cdot 83 + \left(16 a^{2} + 62 a + 32\right)\cdot 83^{2} + \left(14 a^{2} + a + 28\right)\cdot 83^{3} + \left(70 a^{2} + 67 a + 57\right)\cdot 83^{4} +O(83^{5})$$ $r_{ 6 }$ $=$ $$53 a^{2} + 63 a + 23 + \left(20 a^{2} + a + 41\right)\cdot 83 + \left(29 a^{2} + 57 a + 58\right)\cdot 83^{2} + 70 a\cdot 83^{3} + \left(3 a^{2} + 25 a + 6\right)\cdot 83^{4} +O(83^{5})$$ $r_{ 7 }$ $=$ $$52 a^{2} + 65 a + 21 + \left(20 a^{2} + 72 a + 41\right)\cdot 83 + \left(43 a^{2} + a + 3\right)\cdot 83^{2} + \left(18 a^{2} + 62 a + 37\right)\cdot 83^{3} + \left(33 a^{2} + 75 a + 66\right)\cdot 83^{4} +O(83^{5})$$ $r_{ 8 }$ $=$ $$13 a^{2} + 20 a + 26 + \left(81 a^{2} + 42 a + 79\right)\cdot 83 + \left(31 a^{2} + 77 a + 63\right)\cdot 83^{2} + \left(66 a^{2} + 58 a + 49\right)\cdot 83^{3} + \left(76 a^{2} + 72 a + 70\right)\cdot 83^{4} +O(83^{5})$$ $r_{ 9 }$ $=$ $$17 a^{2} + 34 + \left(64 a^{2} + 39 a + 45\right)\cdot 83 + \left(21 a^{2} + 31 a + 43\right)\cdot 83^{2} + \left(16 a^{2} + 36 a + 32\right)\cdot 83^{3} + \left(3 a^{2} + 67 a + 6\right)\cdot 83^{4} +O(83^{5})$$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $1$ $1$ $3$ $(1,7,3)(2,5,4)(6,8,9)$ $\zeta_{9}^{3}$ $1$ $3$ $(1,3,7)(2,4,5)(6,9,8)$ $-\zeta_{9}^{3} - 1$ $1$ $9$ $(1,2,8,7,5,9,3,4,6)$ $\zeta_{9}$ $1$ $9$ $(1,8,5,3,6,2,7,9,4)$ $\zeta_{9}^{2}$ $1$ $9$ $(1,5,6,7,4,8,3,2,9)$ $\zeta_{9}^{4}$ $1$ $9$ $(1,9,2,3,8,4,7,6,5)$ $\zeta_{9}^{5}$ $1$ $9$ $(1,4,9,7,2,6,3,5,8)$ $-\zeta_{9}^{4} - \zeta_{9}$ $1$ $9$ $(1,6,4,3,9,5,7,8,2)$ $-\zeta_{9}^{5} - \zeta_{9}^{2}$

The blue line marks the conjugacy class containing complex conjugation.