# Properties

 Label 2.52.6t5.b.a Dimension $2$ Group $S_3\times C_3$ Conductor $52$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$52$$$$\medspace = 2^{2} \cdot 13$$ Artin stem field: Galois closure of 6.0.10816.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.52.6t1.b.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.676.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{4} - 2x^{3} + 2x + 1$$ x^6 - x^4 - 2*x^3 + 2*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{2} + 29x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$9 a + 19 + \left(14 a + 8\right)\cdot 31 + \left(17 a + 2\right)\cdot 31^{2} + \left(25 a + 4\right)\cdot 31^{3} + \left(23 a + 25\right)\cdot 31^{4} +O(31^{5})$$ 9*a + 19 + (14*a + 8)*31 + (17*a + 2)*31^2 + (25*a + 4)*31^3 + (23*a + 25)*31^4+O(31^5) $r_{ 2 }$ $=$ $$6 a + 21 + \left(23 a + 13\right)\cdot 31 + \left(18 a + 21\right)\cdot 31^{2} + \left(26 a + 15\right)\cdot 31^{3} + \left(25 a + 21\right)\cdot 31^{4} +O(31^{5})$$ 6*a + 21 + (23*a + 13)*31 + (18*a + 21)*31^2 + (26*a + 15)*31^3 + (25*a + 21)*31^4+O(31^5) $r_{ 3 }$ $=$ $$25 a + 2 + \left(7 a + 23\right)\cdot 31 + \left(12 a + 4\right)\cdot 31^{2} + \left(4 a + 19\right)\cdot 31^{3} + \left(5 a + 15\right)\cdot 31^{4} +O(31^{5})$$ 25*a + 2 + (7*a + 23)*31 + (12*a + 4)*31^2 + (4*a + 19)*31^3 + (5*a + 15)*31^4+O(31^5) $r_{ 4 }$ $=$ $$11 a + 27 + \left(9 a + 5\right)\cdot 31 + \left(14 a + 11\right)\cdot 31^{2} + \left(20 a + 10\right)\cdot 31^{3} + \left(28 a + 4\right)\cdot 31^{4} +O(31^{5})$$ 11*a + 27 + (9*a + 5)*31 + (14*a + 11)*31^2 + (20*a + 10)*31^3 + (28*a + 4)*31^4+O(31^5) $r_{ 5 }$ $=$ $$22 a + 6 + \left(16 a + 28\right)\cdot 31 + \left(13 a + 22\right)\cdot 31^{2} + \left(5 a + 6\right)\cdot 31^{3} + \left(7 a + 16\right)\cdot 31^{4} +O(31^{5})$$ 22*a + 6 + (16*a + 28)*31 + (13*a + 22)*31^2 + (5*a + 6)*31^3 + (7*a + 16)*31^4+O(31^5) $r_{ 6 }$ $=$ $$20 a + 18 + \left(21 a + 13\right)\cdot 31 + \left(16 a + 30\right)\cdot 31^{2} + \left(10 a + 5\right)\cdot 31^{3} + \left(2 a + 10\right)\cdot 31^{4} +O(31^{5})$$ 20*a + 18 + (21*a + 13)*31 + (16*a + 30)*31^2 + (10*a + 5)*31^3 + (2*a + 10)*31^4+O(31^5)

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,5)(2,3)(4,6)$ $0$ $1$ $3$ $(1,2,6)(3,4,5)$ $2 \zeta_{3}$ $1$ $3$ $(1,6,2)(3,5,4)$ $-2 \zeta_{3} - 2$ $2$ $3$ $(3,4,5)$ $\zeta_{3} + 1$ $2$ $3$ $(3,5,4)$ $-\zeta_{3}$ $2$ $3$ $(1,2,6)(3,5,4)$ $-1$ $3$ $6$ $(1,4,2,5,6,3)$ $0$ $3$ $6$ $(1,3,6,5,2,4)$ $0$

The blue line marks the conjugacy class containing complex conjugation.