# Properties

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

# Related objects

## Basic invariants

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

## Defining polynomial

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

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

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

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

## 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

 Size Order Action 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.