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:
| $r_{ 1 }$ |
$=$ |
$ 28 + 20\cdot 53 + 21\cdot 53^{2} + 52\cdot 53^{3} + 41\cdot 53^{4} + 5\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 46 + \left(6 a + 31\right)\cdot 53 + \left(42 a + 37\right)\cdot 53^{2} + \left(27 a + 12\right)\cdot 53^{3} + \left(15 a + 30\right)\cdot 53^{4} + \left(22 a + 28\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 a + 43 + \left(34 a + 44\right)\cdot 53 + \left(52 a + 40\right)\cdot 53^{2} + \left(5 a + 20\right)\cdot 53^{3} + \left(18 a + 10\right)\cdot 53^{4} + \left(7 a + 9\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 a + 24 + \left(46 a + 36\right)\cdot 53 + \left(10 a + 40\right)\cdot 53^{2} + \left(25 a + 28\right)\cdot 53^{3} + \left(37 a + 11\right)\cdot 53^{4} + \left(30 a + 49\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 a + 1 + \left(35 a + 21\right)\cdot 53 + \left(46 a + 23\right)\cdot 53^{2} + \left(5 a + 45\right)\cdot 53^{3} + \left(22 a + 42\right)\cdot 53^{4} + \left(5 a + 52\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 13 a + 44 + \left(18 a + 37\right)\cdot 53 + 4\cdot 53^{2} + \left(47 a + 45\right)\cdot 53^{3} + \left(34 a + 23\right)\cdot 53^{4} + \left(45 a + 20\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 3 a + 42 + \left(17 a + 8\right)\cdot 53 + \left(6 a + 15\right)\cdot 53^{2} + \left(47 a + 22\right)\cdot 53^{3} + \left(30 a + 19\right)\cdot 53^{4} + \left(47 a + 52\right)\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 38 + 10\cdot 53 + 28\cdot 53^{2} + 37\cdot 53^{3} + 31\cdot 53^{4} + 46\cdot 53^{5} +O\left(53^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,3)(4,5)(6,8)$ |
| $(1,5,8,4)(2,6,7,3)$ |
| $(1,5,6)(3,8,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $12$ | $2$ | $(1,3)(4,5)(6,8)$ | $0$ |
| $8$ | $3$ | $(1,4,2)(5,7,8)$ | $-1$ |
| $6$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $8$ | $6$ | $(1,7,4,8,2,5)(3,6)$ | $1$ |
| $6$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ | $8$ | $(1,4,3,2,8,5,6,7)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
