# Properties

 Label 2.273.6t5.c.a Dimension $2$ Group $S_3\times C_3$ Conductor $273$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$273$$$$\medspace = 3 \cdot 7 \cdot 13$$ Artin stem field: Galois closure of 6.0.223587.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.273.6t1.d.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.24843.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 3x^{5} + 4x^{4} - 5x^{2} + 3x + 3$$ x^6 - 3*x^5 + 4*x^4 - 5*x^2 + 3*x + 3 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$9 a + 2 + \left(a + 4\right)\cdot 11 + 3\cdot 11^{2} + \left(6 a + 7\right)\cdot 11^{3} + \left(9 a + 2\right)\cdot 11^{4} + \left(6 a + 4\right)\cdot 11^{5} +O(11^{6})$$ 9*a + 2 + (a + 4)*11 + 3*11^2 + (6*a + 7)*11^3 + (9*a + 2)*11^4 + (6*a + 4)*11^5+O(11^6) $r_{ 2 }$ $=$ $$7 a + 8 + \left(7 a + 2\right)\cdot 11 + \left(6 a + 3\right)\cdot 11^{2} + \left(10 a + 1\right)\cdot 11^{3} + \left(a + 2\right)\cdot 11^{4} + \left(10 a + 3\right)\cdot 11^{5} +O(11^{6})$$ 7*a + 8 + (7*a + 2)*11 + (6*a + 3)*11^2 + (10*a + 1)*11^3 + (a + 2)*11^4 + (10*a + 3)*11^5+O(11^6) $r_{ 3 }$ $=$ $$10 a + \left(2 a + 9\right)\cdot 11 + \left(7 a + 9\right)\cdot 11^{2} + \left(8 a + 2\right)\cdot 11^{3} + \left(9 a + 9\right)\cdot 11^{4} + \left(3 a + 10\right)\cdot 11^{5} +O(11^{6})$$ 10*a + (2*a + 9)*11 + (7*a + 9)*11^2 + (8*a + 2)*11^3 + (9*a + 9)*11^4 + (3*a + 10)*11^5+O(11^6) $r_{ 4 }$ $=$ $$a + 7 + \left(8 a + 10\right)\cdot 11 + \left(3 a + 2\right)\cdot 11^{2} + \left(2 a + 8\right)\cdot 11^{3} + \left(a + 6\right)\cdot 11^{4} + \left(7 a + 5\right)\cdot 11^{5} +O(11^{6})$$ a + 7 + (8*a + 10)*11 + (3*a + 2)*11^2 + (2*a + 8)*11^3 + (a + 6)*11^4 + (7*a + 5)*11^5+O(11^6) $r_{ 5 }$ $=$ $$2 a + 5 + \left(9 a + 2\right)\cdot 11 + \left(10 a + 2\right)\cdot 11^{2} + \left(4 a + 9\right)\cdot 11^{3} + \left(a + 1\right)\cdot 11^{4} + 4 a\cdot 11^{5} +O(11^{6})$$ 2*a + 5 + (9*a + 2)*11 + (10*a + 2)*11^2 + (4*a + 9)*11^3 + (a + 1)*11^4 + 4*a*11^5+O(11^6) $r_{ 6 }$ $=$ $$4 a + 3 + \left(3 a + 4\right)\cdot 11 + 4 a\cdot 11^{2} + 4\cdot 11^{3} + \left(9 a + 10\right)\cdot 11^{4} + 8\cdot 11^{5} +O(11^{6})$$ 4*a + 3 + (3*a + 4)*11 + 4*a*11^2 + 4*11^3 + (9*a + 10)*11^4 + 8*11^5+O(11^6)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.