Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 36\cdot 53^{2} + 41\cdot 53^{3} + 14\cdot 53^{4} + 29\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 44\cdot 53 + 5\cdot 53^{2} + 42\cdot 53^{3} + 11\cdot 53^{4} + 44\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 a + 12 + \left(39 a + 20\right)\cdot 53 + \left(6 a + 26\right)\cdot 53^{2} + 5 a\cdot 53^{3} + \left(2 a + 24\right)\cdot 53^{4} + \left(45 a + 16\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 a + 15 + \left(9 a + 7\right)\cdot 53 + \left(42 a + 19\right)\cdot 53^{2} + \left(34 a + 8\right)\cdot 53^{3} + \left(43 a + 39\right)\cdot 53^{4} + \left(30 a + 35\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 a + 16 + \left(43 a + 6\right)\cdot 53 + \left(10 a + 19\right)\cdot 53^{2} + \left(18 a + 52\right)\cdot 53^{3} + \left(9 a + 19\right)\cdot 53^{4} + \left(22 a + 9\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a + 21 + \left(17 a + 52\right)\cdot 53 + \left(7 a + 39\right)\cdot 53^{2} + \left(47 a + 41\right)\cdot 53^{3} + \left(27 a + 31\right)\cdot 53^{4} + \left(48 a + 43\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 36 a + 36 + \left(35 a + 51\right)\cdot 53 + \left(45 a + 51\right)\cdot 53^{2} + \left(5 a + 10\right)\cdot 53^{3} + \left(25 a + 43\right)\cdot 53^{4} + \left(4 a + 50\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 9 a + 29 + \left(13 a + 29\right)\cdot 53 + \left(46 a + 13\right)\cdot 53^{2} + \left(47 a + 14\right)\cdot 53^{3} + \left(50 a + 27\right)\cdot 53^{4} + \left(7 a + 35\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,5,6)(4,8,7)$ |
| $(1,8,2,3)(4,7,6,5)$ |
| $(1,2)(3,8)(4,6)(5,7)$ |
| $(3,8)(4,5)(6,7)$ |
| $(1,4,2,6)(3,7,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $-2$ |
| $12$ | $2$ | $(3,8)(4,5)(6,7)$ | $0$ |
| $8$ | $3$ | $(1,7,3)(2,5,8)$ | $-1$ |
| $6$ | $4$ | $(1,8,2,3)(4,7,6,5)$ | $0$ |
| $8$ | $6$ | $(1,2)(3,7,6,8,5,4)$ | $1$ |
| $6$ | $8$ | $(1,4,3,5,2,6,8,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,6,3,7,2,4,8,5)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
