# Properties

 Label 1.273.6t1.h.b 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 6.0.24069811311.2 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{273}(179,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 34x^{4} - 9x^{3} + 786x^{2} + 738x + 9099$$ x^6 - x^5 + 34*x^4 - 9*x^3 + 786*x^2 + 738*x + 9099 .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$16 a + 5 + \left(18 a + 9\right)\cdot 19 + \left(9 a + 9\right)\cdot 19^{2} + \left(10 a + 13\right)\cdot 19^{3} + \left(13 a + 4\right)\cdot 19^{4} +O(19^{5})$$ 16*a + 5 + (18*a + 9)*19 + (9*a + 9)*19^2 + (10*a + 13)*19^3 + (13*a + 4)*19^4+O(19^5) $r_{ 2 }$ $=$ $$8 a + 6 + \left(13 a + 13\right)\cdot 19 + \left(a + 10\right)\cdot 19^{2} + \left(9 a + 4\right)\cdot 19^{3} + \left(11 a + 10\right)\cdot 19^{4} +O(19^{5})$$ 8*a + 6 + (13*a + 13)*19 + (a + 10)*19^2 + (9*a + 4)*19^3 + (11*a + 10)*19^4+O(19^5) $r_{ 3 }$ $=$ $$3 a + 2 + 12\cdot 19 + 9 a\cdot 19^{2} + \left(8 a + 14\right)\cdot 19^{3} + \left(5 a + 7\right)\cdot 19^{4} +O(19^{5})$$ 3*a + 2 + 12*19 + 9*a*19^2 + (8*a + 14)*19^3 + (5*a + 7)*19^4+O(19^5) $r_{ 4 }$ $=$ $$18 a + 16 + \left(13 a + 3\right)\cdot 19 + \left(18 a + 16\right)\cdot 19^{2} + \left(18 a + 15\right)\cdot 19^{3} + \left(15 a + 2\right)\cdot 19^{4} +O(19^{5})$$ 18*a + 16 + (13*a + 3)*19 + (18*a + 16)*19^2 + (18*a + 15)*19^3 + (15*a + 2)*19^4+O(19^5) $r_{ 5 }$ $=$ $$11 a + 14 + \left(5 a + 18\right)\cdot 19 + \left(17 a + 17\right)\cdot 19^{2} + \left(9 a + 11\right)\cdot 19^{3} + \left(7 a + 12\right)\cdot 19^{4} +O(19^{5})$$ 11*a + 14 + (5*a + 18)*19 + (17*a + 17)*19^2 + (9*a + 11)*19^3 + (7*a + 12)*19^4+O(19^5) $r_{ 6 }$ $=$ $$a + 15 + \left(5 a + 18\right)\cdot 19 + 19^{2} + 16\cdot 19^{3} + \left(3 a + 18\right)\cdot 19^{4} +O(19^{5})$$ a + 15 + (5*a + 18)*19 + 19^2 + 16*19^3 + (3*a + 18)*19^4+O(19^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.