# Properties

 Label 1.104.6t1.b.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: Galois closure of 6.0.190102016.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{104}(43,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 26x^{4} + 104x^{2} + 104$$ x^6 + 26*x^4 + 104*x^2 + 104 .

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

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

Roots:
 $r_{ 1 }$ $=$ $$52 a + 2 + 12 a\cdot 53 + \left(22 a + 15\right)\cdot 53^{2} + \left(46 a + 24\right)\cdot 53^{3} + \left(13 a + 48\right)\cdot 53^{4} +O(53^{5})$$ 52*a + 2 + 12*a*53 + (22*a + 15)*53^2 + (46*a + 24)*53^3 + (13*a + 48)*53^4+O(53^5) $r_{ 2 }$ $=$ $$19 a + 15 + \left(15 a + 5\right)\cdot 53 + \left(21 a + 18\right)\cdot 53^{2} + \left(28 a + 33\right)\cdot 53^{3} + \left(5 a + 29\right)\cdot 53^{4} +O(53^{5})$$ 19*a + 15 + (15*a + 5)*53 + (21*a + 18)*53^2 + (28*a + 33)*53^3 + (5*a + 29)*53^4+O(53^5) $r_{ 3 }$ $=$ $$46 a + 14 + \left(28 a + 18\right)\cdot 53 + \left(30 a + 6\right)\cdot 53^{2} + \left(51 a + 18\right)\cdot 53^{3} + \left(30 a + 43\right)\cdot 53^{4} +O(53^{5})$$ 46*a + 14 + (28*a + 18)*53 + (30*a + 6)*53^2 + (51*a + 18)*53^3 + (30*a + 43)*53^4+O(53^5) $r_{ 4 }$ $=$ $$a + 51 + \left(40 a + 52\right)\cdot 53 + \left(30 a + 37\right)\cdot 53^{2} + \left(6 a + 28\right)\cdot 53^{3} + \left(39 a + 4\right)\cdot 53^{4} +O(53^{5})$$ a + 51 + (40*a + 52)*53 + (30*a + 37)*53^2 + (6*a + 28)*53^3 + (39*a + 4)*53^4+O(53^5) $r_{ 5 }$ $=$ $$34 a + 38 + \left(37 a + 47\right)\cdot 53 + \left(31 a + 34\right)\cdot 53^{2} + \left(24 a + 19\right)\cdot 53^{3} + \left(47 a + 23\right)\cdot 53^{4} +O(53^{5})$$ 34*a + 38 + (37*a + 47)*53 + (31*a + 34)*53^2 + (24*a + 19)*53^3 + (47*a + 23)*53^4+O(53^5) $r_{ 6 }$ $=$ $$7 a + 39 + \left(24 a + 34\right)\cdot 53 + \left(22 a + 46\right)\cdot 53^{2} + \left(a + 34\right)\cdot 53^{3} + \left(22 a + 9\right)\cdot 53^{4} +O(53^{5})$$ 7*a + 39 + (24*a + 34)*53 + (22*a + 46)*53^2 + (a + 34)*53^3 + (22*a + 9)*53^4+O(53^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.