# Properties

 Label 1.252.6t1.j.b Dimension $1$ Group $C_6$ Conductor $252$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$252$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 7$$ Artin field: 6.0.432081216.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{252}(167,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 42 x^{4} + 441 x^{2} + 1029$$  .

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $$x^{2} + 49 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$31 a + 44 + \left(36 a + 21\right)\cdot 53 + \left(51 a + 47\right)\cdot 53^{2} + \left(22 a + 32\right)\cdot 53^{3} + \left(32 a + 52\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 2 }$ $=$ $$25 a + 3 + \left(15 a + 8\right)\cdot 53 + \left(49 a + 15\right)\cdot 53^{2} + \left(22 a + 5\right)\cdot 53^{3} + \left(38 a + 14\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 3 }$ $=$ $$47 a + 12 + \left(31 a + 39\right)\cdot 53 + \left(50 a + 20\right)\cdot 53^{2} + \left(52 a + 25\right)\cdot 53^{3} + \left(5 a + 14\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 4 }$ $=$ $$22 a + 9 + \left(16 a + 31\right)\cdot 53 + \left(a + 5\right)\cdot 53^{2} + \left(30 a + 20\right)\cdot 53^{3} + 20 a\cdot 53^{4} +O(53^{5})$$ $r_{ 5 }$ $=$ $$28 a + 50 + \left(37 a + 44\right)\cdot 53 + \left(3 a + 37\right)\cdot 53^{2} + \left(30 a + 47\right)\cdot 53^{3} + \left(14 a + 38\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 6 }$ $=$ $$6 a + 41 + \left(21 a + 13\right)\cdot 53 + \left(2 a + 32\right)\cdot 53^{2} + 27\cdot 53^{3} + \left(47 a + 38\right)\cdot 53^{4} +O(53^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.