Properties

Label 1.19.9t1.a.f
Dimension $1$
Group $C_9$
Conductor $19$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_9$
Conductor: \(19\)
Artin field: \(\Q(\zeta_{19})^+\)
Galois orbit size: $6$
Smallest permutation container: $C_9$
Parity: even
Dirichlet character: \(\chi_{19}(17,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{9} - x^{8} - 8 x^{7} + 7 x^{6} + 21 x^{5} - 15 x^{4} - 20 x^{3} + 10 x^{2} + 5 x - 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{3} + 2 x + 9\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a^{2} + 4 a + \left(a^{2} + 6 a + 2\right)\cdot 11 + \left(4 a^{2} + 3 a + 4\right)\cdot 11^{2} + \left(a^{2} + 3 a + 5\right)\cdot 11^{3} + \left(4 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(8 a^{2} + 8 a + 6\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 4 a^{2} + 2 a + 8 + \left(2 a^{2} + 4 a + 9\right)\cdot 11 + \left(5 a^{2} + 9 a + 9\right)\cdot 11^{2} + \left(9 a^{2} + a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 2 a + 9\right)\cdot 11^{4} + \left(9 a^{2} + 6 a + 9\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 2 a^{2} + 8 a + 9 + \left(7 a^{2} + 5 a + 8\right)\cdot 11 + \left(a^{2} + 8 a + 8\right)\cdot 11^{2} + \left(9 a^{2} + 4 a + 9\right)\cdot 11^{3} + \left(6 a^{2} + 4 a + 7\right)\cdot 11^{4} + \left(2 a^{2} + 7 a + 7\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 9 a^{2} + 6 a + 8 + \left(a^{2} + 2 a + 2\right)\cdot 11 + \left(5 a^{2} + 9\right)\cdot 11^{2} + \left(6 a^{2} + 3 a + 4\right)\cdot 11^{3} + \left(6 a^{2} + 9 a + 6\right)\cdot 11^{4} + \left(2 a^{2} + 8 a + 6\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 5 a^{2} + a + 2 + \left(a^{2} + a + 1\right)\cdot 11 + \left(4 a^{2} + 4 a + 1\right)\cdot 11^{2} + \left(3 a^{2} + 4 a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 4 a + 1\right)\cdot 11^{4} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 10 a^{2} + a + 2 + \left(7 a^{2} + 2 a + 7\right)\cdot 11 + \left(a^{2} + 7 a + 4\right)\cdot 11^{2} + \left(3 a^{2} + 4 a\right)\cdot 11^{3} + \left(3 a + 9\right)\cdot 11^{4} + \left(4 a + 6\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 7 a^{2} + \left(7 a^{2} + 8 a + 7\right)\cdot 11 + \left(4 a^{2} + 2 a + 4\right)\cdot 11^{2} + \left(5 a^{2} + 9 a + 2\right)\cdot 11^{3} + \left(6 a^{2} + a + 1\right)\cdot 11^{4} + \left(5 a^{2} + 3 a\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 7 a^{2} + 10 a + \left(6 a^{2} + 8 a + 2\right)\cdot 11 + \left(a^{2} + 9 a + 4\right)\cdot 11^{2} + \left(4 a + 6\right)\cdot 11^{3} + \left(10 a^{2} + 10 a + 9\right)\cdot 11^{4} + \left(8 a^{2} + a\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 9 }$ $=$ \( 8 a^{2} + a + 5 + \left(7 a^{2} + 5 a + 3\right)\cdot 11 + \left(4 a^{2} + 9 a + 8\right)\cdot 11^{2} + \left(5 a^{2} + 7 a + 9\right)\cdot 11^{3} + \left(5 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(7 a^{2} + 5 a + 9\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.