# Properties

 Label 2.1352.6t5.a Dimension $2$ Group $S_3\times C_3$ Conductor $1352$ Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$1352$$$$\medspace = 2^{3} \cdot 13^{2}$$ Artin number field: Galois closure of 6.0.190102016.3 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.104.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $$x^{2} + 49x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$52 a + 3 + \left(a + 12\right)\cdot 53 + \left(34 a + 30\right)\cdot 53^{2} + \left(8 a + 50\right)\cdot 53^{3} + \left(39 a + 24\right)\cdot 53^{4} + \left(41 a + 51\right)\cdot 53^{5} +O(53^{6})$$ 52*a + 3 + (a + 12)*53 + (34*a + 30)*53^2 + (8*a + 50)*53^3 + (39*a + 24)*53^4 + (41*a + 51)*53^5+O(53^6) $r_{ 2 }$ $=$ $$13 a + 24 + \left(12 a + 23\right)\cdot 53 + \left(41 a + 11\right)\cdot 53^{2} + \left(51 a + 31\right)\cdot 53^{3} + \left(48 a + 32\right)\cdot 53^{4} + \left(47 a + 19\right)\cdot 53^{5} +O(53^{6})$$ 13*a + 24 + (12*a + 23)*53 + (41*a + 11)*53^2 + (51*a + 31)*53^3 + (48*a + 32)*53^4 + (47*a + 19)*53^5+O(53^6) $r_{ 3 }$ $=$ $$14 a + 28 + \left(10 a + 34\right)\cdot 53 + \left(7 a + 17\right)\cdot 53^{2} + \left(43 a + 17\right)\cdot 53^{3} + \left(9 a + 10\right)\cdot 53^{4} + \left(6 a + 51\right)\cdot 53^{5} +O(53^{6})$$ 14*a + 28 + (10*a + 34)*53 + (7*a + 17)*53^2 + (43*a + 17)*53^3 + (9*a + 10)*53^4 + (6*a + 51)*53^5+O(53^6) $r_{ 4 }$ $=$ $$39 a + 31 + \left(42 a + 8\right)\cdot 53 + \left(45 a + 36\right)\cdot 53^{2} + \left(9 a + 23\right)\cdot 53^{3} + \left(43 a + 6\right)\cdot 53^{4} + \left(46 a + 13\right)\cdot 53^{5} +O(53^{6})$$ 39*a + 31 + (42*a + 8)*53 + (45*a + 36)*53^2 + (9*a + 23)*53^3 + (43*a + 6)*53^4 + (46*a + 13)*53^5+O(53^6) $r_{ 5 }$ $=$ $$a + 52 + \left(51 a + 20\right)\cdot 53 + \left(18 a + 5\right)\cdot 53^{2} + \left(44 a + 51\right)\cdot 53^{3} + \left(13 a + 13\right)\cdot 53^{4} + \left(11 a + 20\right)\cdot 53^{5} +O(53^{6})$$ a + 52 + (51*a + 20)*53 + (18*a + 5)*53^2 + (44*a + 51)*53^3 + (13*a + 13)*53^4 + (11*a + 20)*53^5+O(53^6) $r_{ 6 }$ $=$ $$40 a + 23 + \left(40 a + 6\right)\cdot 53 + \left(11 a + 5\right)\cdot 53^{2} + \left(a + 38\right)\cdot 53^{3} + \left(4 a + 17\right)\cdot 53^{4} + \left(5 a + 3\right)\cdot 53^{5} +O(53^{6})$$ 40*a + 23 + (40*a + 6)*53 + (11*a + 5)*53^2 + (a + 38)*53^3 + (4*a + 17)*53^4 + (5*a + 3)*53^5+O(53^6)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $2$ $2$ $3$ $2$ $(1,4)(2,3)(5,6)$ $0$ $0$ $1$ $3$ $(1,3,6)(2,5,4)$ $2 \zeta_{3}$ $-2 \zeta_{3} - 2$ $1$ $3$ $(1,6,3)(2,4,5)$ $-2 \zeta_{3} - 2$ $2 \zeta_{3}$ $2$ $3$ $(1,3,6)(2,4,5)$ $-1$ $-1$ $2$ $3$ $(2,4,5)$ $-\zeta_{3}$ $\zeta_{3} + 1$ $2$ $3$ $(2,5,4)$ $\zeta_{3} + 1$ $-\zeta_{3}$ $3$ $6$ $(1,5,3,4,6,2)$ $0$ $0$ $3$ $6$ $(1,2,6,4,3,5)$ $0$ $0$
The blue line marks the conjugacy class containing complex conjugation.