Basic invariants
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:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 5 + \left(39 a + 34\right)\cdot 53 + \left(14 a + 30\right)\cdot 53^{2} + \left(2 a + 2\right)\cdot 53^{3} + \left(47 a + 43\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a + 27 + \left(39 a + 22\right)\cdot 53 + \left(14 a + 29\right)\cdot 53^{2} + \left(2 a + 46\right)\cdot 53^{3} + \left(47 a + 40\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 a + 37 + \left(39 a + 3\right)\cdot 53 + \left(14 a + 16\right)\cdot 53^{2} + \left(2 a + 12\right)\cdot 53^{3} + \left(47 a + 8\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 a + 35 + \left(13 a + 17\right)\cdot 53 + \left(38 a + 49\right)\cdot 53^{2} + \left(50 a + 40\right)\cdot 53^{3} + \left(5 a + 14\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 a + 13 + \left(13 a + 29\right)\cdot 53 + \left(38 a + 50\right)\cdot 53^{2} + \left(50 a + 49\right)\cdot 53^{3} + \left(5 a + 16\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 a + 45 + \left(13 a + 51\right)\cdot 53 + \left(38 a + 35\right)\cdot 53^{2} + \left(50 a + 6\right)\cdot 53^{3} + \left(5 a + 35\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,4)(3,6)$ |
| $(1,2,3)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,4)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,3,5,2,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,2,5,3,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
