# Properties

 Label 1.104.6t1.a.b Dimension $1$ Group $C_6$ Conductor $104$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$104$$$$\medspace = 2^{3} \cdot 13$$ Artin field: 6.6.14623232.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{104}(61,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2 x^{5} - 13 x^{4} + 14 x^{3} + 26 x^{2} - 28 x + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $$x^{2} + 49 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$52 a + 16 + \left(25 a + 15\right)\cdot 53 + \left(29 a + 37\right)\cdot 53^{2} + \left(34 a + 49\right)\cdot 53^{3} + \left(38 a + 19\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 2 }$ $=$ $$52 a + 35 + \left(25 a + 29\right)\cdot 53 + \left(29 a + 52\right)\cdot 53^{2} + \left(34 a + 37\right)\cdot 53^{3} + \left(38 a + 50\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 3 }$ $=$ $$a + 5 + \left(27 a + 35\right)\cdot 53 + \left(23 a + 49\right)\cdot 53^{2} + \left(18 a + 16\right)\cdot 53^{3} + \left(14 a + 28\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 4 }$ $=$ $$52 a + 9 + \left(25 a + 36\right)\cdot 53 + \left(29 a + 10\right)\cdot 53^{2} + \left(34 a + 14\right)\cdot 53^{3} + \left(38 a + 14\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 5 }$ $=$ $$a + 31 + \left(27 a + 28\right)\cdot 53 + \left(23 a + 38\right)\cdot 53^{2} + \left(18 a + 40\right)\cdot 53^{3} + \left(14 a + 11\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 6 }$ $=$ $$a + 12 + \left(27 a + 14\right)\cdot 53 + \left(23 a + 23\right)\cdot 53^{2} + \left(18 a + 52\right)\cdot 53^{3} + \left(14 a + 33\right)\cdot 53^{4} +O(53^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.