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: | Galois closure of \(\Q\) |
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 $\mathbb{Q}_{ 53 }$:
\( x^{2} + 49x + 2 \)
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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 |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,6,4)(2,5,3)$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,4,6)(2,3,5)$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,6,2,4,5)$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,4,2,6,3)$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ |