# Properties

 Label 1.273.6t1.d.a Dimension $1$ Group $C_6$ Conductor $273$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$273$$$$\medspace = 3 \cdot 7 \cdot 13$$ Artin field: Galois closure of 6.0.1851523947.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{273}(191,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 31x^{4} + 84x^{3} + 873x^{2} + 810x + 729$$ x^6 - x^5 + 31*x^4 + 84*x^3 + 873*x^2 + 810*x + 729 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

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

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.