# Properties

 Label 1.273.6t1.b.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: 6.6.168488679177.2 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{273}(17,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} - 83 x^{4} + 433 x^{3} - 566 x^{2} - 120 x + 64$$  .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: $$x^{2} + 21 x + 5$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.