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:
| $r_{ 1 }$ |
$=$ |
$ 47 a + 29 + \left(46 a + 45\right)\cdot 53 + \left(25 a + 51\right)\cdot 53^{2} + \left(40 a + 5\right)\cdot 53^{3} + \left(13 a + 1\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 a + 34 + \left(30 a + 47\right)\cdot 53 + \left(25 a + 38\right)\cdot 53^{2} + \left(40 a + 44\right)\cdot 53^{3} + \left(19 a + 43\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 a + 28 + \left(22 a + 38\right)\cdot 53 + \left(27 a + 4\right)\cdot 53^{2} + \left(12 a + 22\right)\cdot 53^{3} + \left(33 a + 29\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 5 + \left(6 a + 27\right)\cdot 53 + \left(27 a + 2\right)\cdot 53^{2} + \left(12 a + 36\right)\cdot 53^{3} + \left(39 a + 15\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 35\cdot 53 + 46\cdot 53^{2} + 35\cdot 53^{3} + 3\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 17\cdot 53 + 14\cdot 53^{2} + 14\cdot 53^{3} + 12\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
