# Properties

 Label 1.152.6t1.c.b Dimension $1$ Group $C_6$ Conductor $152$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$152$$$$\medspace = 2^{3} \cdot 19$$ Artin field: Galois closure of 6.0.66724352.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{152}(83,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} - 5x^{4} + 18x^{3} + 38x^{2} - 152x + 209$$ x^6 - 2*x^5 - 5*x^4 + 18*x^3 + 38*x^2 - 152*x + 209 .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6x + 3$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.