# Properties

 Label 1.133.6t1.a Dimension $1$ Group $C_6$ Conductor $133$ Indicator $0$

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## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$133$$$$\medspace = 7 \cdot 19$$ Artin number field: Galois closure of 6.0.5945113699.2 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Projective image: $C_1$ Projective field: $$\Q$$

## Galois action

### Roots of defining polynomial

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:
 $r_{ 1 }$ $=$ $$22 a + \left(6 a + 3\right)\cdot 53 + \left(3 a + 51\right)\cdot 53^{2} + \left(a + 14\right)\cdot 53^{3} + \left(46 a + 4\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 2 }$ $=$ $$31 a + 35 + \left(46 a + 6\right)\cdot 53 + \left(49 a + 4\right)\cdot 53^{2} + \left(51 a + 16\right)\cdot 53^{3} + \left(6 a + 28\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 3 }$ $=$ $$33 a + 13 + \left(12 a + 25\right)\cdot 53 + \left(36 a + 2\right)\cdot 53^{2} + \left(33 a + 31\right)\cdot 53^{3} + \left(24 a + 7\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 4 }$ $=$ $$20 a + 39 + \left(40 a + 42\right)\cdot 53 + \left(16 a + 28\right)\cdot 53^{2} + \left(19 a + 23\right)\cdot 53^{3} + \left(28 a + 19\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 5 }$ $=$ $$5 a + \left(24 a + 48\right)\cdot 53 + \left(7 a + 6\right)\cdot 53^{2} + \left(50 a + 46\right)\cdot 53^{3} + \left(7 a + 5\right)\cdot 53^{4} +O(53^{5})$$ $r_{ 6 }$ $=$ $$48 a + 20 + \left(28 a + 33\right)\cdot 53 + \left(45 a + 12\right)\cdot 53^{2} + \left(2 a + 27\right)\cdot 53^{3} + \left(45 a + 40\right)\cdot 53^{4} +O(53^{5})$$

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $c2$ $1$ $1$ $()$ $1$ $1$ $1$ $2$ $(1,2)(3,4)(5,6)$ $-1$ $-1$ $1$ $3$ $(1,6,4)(2,5,3)$ $\zeta_{3}$ $-\zeta_{3} - 1$ $1$ $3$ $(1,4,6)(2,3,5)$ $-\zeta_{3} - 1$ $\zeta_{3}$ $1$ $6$ $(1,3,6,2,4,5)$ $\zeta_{3} + 1$ $-\zeta_{3}$ $1$ $6$ $(1,5,4,2,6,3)$ $-\zeta_{3}$ $\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.