# Properties

 Label 1.28.6t1.a.a Dimension $1$ Group $C_6$ Conductor $28$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$28$$$$\medspace = 2^{2} \cdot 7$$ Artin field: 6.0.153664.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{28}(11,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $$x^{2} + 42 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$32 a + 27 + \left(29 a + 22\right)\cdot 43 + \left(10 a + 9\right)\cdot 43^{2} + \left(18 a + 39\right)\cdot 43^{3} + \left(27 a + 16\right)\cdot 43^{4} +O(43^{5})$$ $r_{ 2 }$ $=$ $$10 a + 38 + \left(19 a + 16\right)\cdot 43 + \left(38 a + 33\right)\cdot 43^{2} + \left(15 a + 32\right)\cdot 43^{3} + \left(6 a + 4\right)\cdot 43^{4} +O(43^{5})$$ $r_{ 3 }$ $=$ $$26 a + 30 + \left(35 a + 16\right)\cdot 43 + \left(36 a + 42\right)\cdot 43^{2} + \left(28 a + 3\right)\cdot 43^{3} + \left(4 a + 12\right)\cdot 43^{4} +O(43^{5})$$ $r_{ 4 }$ $=$ $$11 a + 16 + \left(13 a + 20\right)\cdot 43 + \left(32 a + 33\right)\cdot 43^{2} + \left(24 a + 3\right)\cdot 43^{3} + \left(15 a + 26\right)\cdot 43^{4} +O(43^{5})$$ $r_{ 5 }$ $=$ $$33 a + 5 + \left(23 a + 26\right)\cdot 43 + \left(4 a + 9\right)\cdot 43^{2} + \left(27 a + 10\right)\cdot 43^{3} + \left(36 a + 38\right)\cdot 43^{4} +O(43^{5})$$ $r_{ 6 }$ $=$ $$17 a + 13 + \left(7 a + 26\right)\cdot 43 + 6 a\cdot 43^{2} + \left(14 a + 39\right)\cdot 43^{3} + \left(38 a + 30\right)\cdot 43^{4} +O(43^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.