Properties

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

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

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

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

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

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

SizeOrderAction 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.