# Properties

 Label 1.1169.6t1.b.b Dimension $1$ Group $C_6$ Conductor $1169$ Root number not computed Indicator $0$

# Learn more

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$1169$$$$\medspace = 7 \cdot 167$$ Artin field: Galois closure of 6.0.11182568663.3 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{1169}(667,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 121x^{4} - 81x^{3} + 5255x^{2} - 1681x + 81269$$ x^6 - x^5 + 121*x^4 - 81*x^3 + 5255*x^2 - 1681*x + 81269 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12x + 2$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.