Properties

Label 1.199.9t1.a.b
Dimension $1$
Group $C_9$
Conductor $199$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_9$
Conductor: \(199\)
Artin field: 9.9.2459374191553118401.1
Galois orbit size: $6$
Smallest permutation container: $C_9$
Parity: even
Dirichlet character: \(\chi_{199}(43,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{9} - x^{8} - 88 x^{7} + 325 x^{6} + 775 x^{5} - 3447 x^{4} - 1602 x^{3} + 7354 x^{2} - 3333 x - 121 $.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{3} + x + 14 $

Roots:
$r_{ 1 }$ $=$ $ 6 a^{2} + 11 a + 2 + \left(7 a^{2} + 8 a + 6\right)\cdot 17 + \left(3 a^{2} + 13 a + 6\right)\cdot 17^{2} + \left(2 a^{2} + 7 a\right)\cdot 17^{3} + \left(4 a^{2} + 16\right)\cdot 17^{4} + \left(10 a^{2} + 6 a + 11\right)\cdot 17^{5} + \left(2 a^{2} + 12 a + 11\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 16 a^{2} + 15 a + \left(14 a^{2} + 9 a + 3\right)\cdot 17 + \left(6 a^{2} + 14 a + 12\right)\cdot 17^{2} + \left(14 a^{2} + 13 a + 10\right)\cdot 17^{3} + \left(7 a^{2} + 12 a + 15\right)\cdot 17^{4} + \left(6 a^{2} + a + 15\right)\cdot 17^{5} + \left(a^{2} + 6 a + 2\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 11 a^{2} + 8 a + 9 + \left(5 a^{2} + 3 a + 9\right)\cdot 17 + \left(9 a^{2} + 10 a + 11\right)\cdot 17^{2} + \left(15 a^{2} + 10 a + 4\right)\cdot 17^{3} + \left(a^{2} + 11 a\right)\cdot 17^{4} + \left(13 a^{2} + 11 a + 9\right)\cdot 17^{5} + \left(10 a^{2} + 8 a + 6\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 7 a^{2} + 13 a + 12 + \left(8 a^{2} + 11 a + 5\right)\cdot 17 + \left(4 a^{2} + 11 a + 8\right)\cdot 17^{2} + \left(14 a^{2} + 5 a + 9\right)\cdot 17^{3} + \left(2 a^{2} + 10 a + 6\right)\cdot 17^{4} + \left(10 a^{2} + 6 a + 1\right)\cdot 17^{5} + \left(7 a^{2} + 11 a + 10\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 16 a^{2} + 10 a + \left(2 a + 5\right)\cdot 17 + \left(11 a^{2} + 7 a + 9\right)\cdot 17^{2} + \left(11 a^{2} + 5 a + 14\right)\cdot 17^{3} + \left(14 a^{2} + 10 a + 8\right)\cdot 17^{4} + \left(5 a^{2} + 10 a + 15\right)\cdot 17^{5} + \left(8 a^{2} + 3 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 6 }$ $=$ $ a^{2} + 5 a + 10 + \left(10 a^{2} + 15 a + 13\right)\cdot 17 + \left(2 a^{2} + 5 a + 5\right)\cdot 17^{2} + \left(11 a^{2} + 15 a + 6\right)\cdot 17^{3} + \left(3 a + 2\right)\cdot 17^{4} + \left(6 a^{2} + 4 a + 9\right)\cdot 17^{5} + \left(3 a^{2} + 9 a + 6\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 2 a^{2} + 9 a + 2 + \left(a^{2} + 4 a + 5\right)\cdot 17 + \left(16 a^{2} + 12 a + 1\right)\cdot 17^{2} + \left(7 a^{2} + 14 a + 12\right)\cdot 17^{3} + \left(11 a^{2} + 10 a + 6\right)\cdot 17^{4} + \left(4 a^{2} + 4 a + 3\right)\cdot 17^{5} + \left(7 a^{2} + 7 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 10 a^{2} + a + 16 + \left(16 a^{2} + 10 a\right)\cdot 17 + \left(10 a^{2} + 14 a\right)\cdot 17^{2} + \left(3 a^{2} + 10 a + 7\right)\cdot 17^{3} + \left(12 a^{2} + 12 a + 4\right)\cdot 17^{4} + \left(6 a + 11\right)\cdot 17^{5} + \left(11 a^{2} + 12 a + 11\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 9 }$ $=$ $ 16 a^{2} + 13 a + 1 + \left(2 a^{2} + a + 2\right)\cdot 17 + \left(3 a^{2} + 12 a + 13\right)\cdot 17^{2} + \left(4 a^{2} + 2\right)\cdot 17^{3} + \left(12 a^{2} + 12 a + 7\right)\cdot 17^{4} + \left(10 a^{2} + 15 a + 7\right)\cdot 17^{5} + \left(15 a^{2} + 13 a + 15\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.