# Properties

 Label 1.532.6t1.f Dimension $1$ Group $C_6$ Conductor $532$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$532$$$$\medspace = 2^{2} \cdot 7 \cdot 19$$ Artin number field: Galois closure of 6.0.2663410937152.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21 x + 5$$
Roots:
 $r_{ 1 }$ $=$ $$3 a + 20 + \left(9 a + 3\right)\cdot 23 + \left(a + 3\right)\cdot 23^{2} + \left(22 a + 13\right)\cdot 23^{3} + \left(17 a + 4\right)\cdot 23^{4} + \left(6 a + 2\right)\cdot 23^{5} +O(23^{6})$$ $r_{ 2 }$ $=$ $$15 a + 8 + \left(7 a + 11\right)\cdot 23 + \left(a + 2\right)\cdot 23^{2} + \left(2 a + 10\right)\cdot 23^{3} + \left(21 a + 14\right)\cdot 23^{4} + \left(2 a + 7\right)\cdot 23^{5} +O(23^{6})$$ $r_{ 3 }$ $=$ $$14 a + 9 + \left(20 a + 9\right)\cdot 23 + \left(2 a + 7\right)\cdot 23^{2} + \left(14 a + 10\right)\cdot 23^{3} + \left(10 a + 19\right)\cdot 23^{4} + \left(20 a + 7\right)\cdot 23^{5} +O(23^{6})$$ $r_{ 4 }$ $=$ $$20 a + 3 + \left(13 a + 19\right)\cdot 23 + \left(21 a + 19\right)\cdot 23^{2} + 9\cdot 23^{3} + \left(5 a + 18\right)\cdot 23^{4} + \left(16 a + 20\right)\cdot 23^{5} +O(23^{6})$$ $r_{ 5 }$ $=$ $$8 a + 15 + \left(15 a + 11\right)\cdot 23 + \left(21 a + 20\right)\cdot 23^{2} + \left(20 a + 12\right)\cdot 23^{3} + \left(a + 8\right)\cdot 23^{4} + \left(20 a + 15\right)\cdot 23^{5} +O(23^{6})$$ $r_{ 6 }$ $=$ $$9 a + 14 + \left(2 a + 13\right)\cdot 23 + \left(20 a + 15\right)\cdot 23^{2} + \left(8 a + 12\right)\cdot 23^{3} + \left(12 a + 3\right)\cdot 23^{4} + \left(2 a + 15\right)\cdot 23^{5} +O(23^{6})$$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,4)(2,5)(3,6)$ $-1$ $-1$ $1$ $3$ $(1,6,2)(3,5,4)$ $\zeta_{3}$ $-\zeta_{3} - 1$ $1$ $3$ $(1,2,6)(3,4,5)$ $-\zeta_{3} - 1$ $\zeta_{3}$ $1$ $6$ $(1,5,6,4,2,3)$ $\zeta_{3} + 1$ $-\zeta_{3}$ $1$ $6$ $(1,3,2,4,6,5)$ $-\zeta_{3}$ $\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.