# Properties

 Label 1.273.6t1.g.b Dimension $1$ Group $C_6$ Conductor $273$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$273$$$$\medspace = 3 \cdot 7 \cdot 13$$ Artin field: Galois closure of 6.0.24069811311.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{273}(95,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 34x^{4} + 82x^{3} - 215x^{2} - 81x + 4276$$ x^6 - x^5 + 34*x^4 + 82*x^3 - 215*x^2 - 81*x + 4276 .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16x + 3$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.