Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 16 + \left(23 a + 32\right)\cdot 53 + \left(37 a + 33\right)\cdot 53^{2} + \left(26 a + 33\right)\cdot 53^{3} + \left(7 a + 18\right)\cdot 53^{4} + \left(28 a + 52\right)\cdot 53^{5} + \left(3 a + 10\right)\cdot 53^{6} + \left(a + 29\right)\cdot 53^{7} + \left(5 a + 46\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 a + 39 + \left(47 a + 33\right)\cdot 53 + \left(14 a + 20\right)\cdot 53^{2} + \left(46 a + 47\right)\cdot 53^{3} + \left(18 a + 11\right)\cdot 53^{4} + \left(27 a + 34\right)\cdot 53^{5} + \left(23 a + 19\right)\cdot 53^{6} + \left(20 a + 50\right)\cdot 53^{7} + \left(48 a + 45\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 a + 15 + \left(5 a + 19\right)\cdot 53 + \left(38 a + 32\right)\cdot 53^{2} + \left(6 a + 5\right)\cdot 53^{3} + \left(34 a + 41\right)\cdot 53^{4} + \left(25 a + 18\right)\cdot 53^{5} + \left(29 a + 33\right)\cdot 53^{6} + \left(32 a + 2\right)\cdot 53^{7} + \left(4 a + 7\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 40 + \left(23 a + 5\right)\cdot 53 + \left(37 a + 52\right)\cdot 53^{2} + \left(26 a + 2\right)\cdot 53^{3} + \left(7 a + 31\right)\cdot 53^{4} + \left(28 a + 1\right)\cdot 53^{5} + \left(3 a + 3\right)\cdot 53^{6} + \left(a + 23\right)\cdot 53^{7} + \left(5 a + 40\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 38 + \left(29 a + 20\right)\cdot 53 + \left(15 a + 19\right)\cdot 53^{2} + \left(26 a + 19\right)\cdot 53^{3} + \left(45 a + 34\right)\cdot 53^{4} + 24 a\cdot 53^{5} + \left(49 a + 42\right)\cdot 53^{6} + \left(51 a + 23\right)\cdot 53^{7} + \left(47 a + 6\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 45 + 42\cdot 53 + 10\cdot 53^{2} + 24\cdot 53^{3} + 15\cdot 53^{4} + 8\cdot 53^{5} + 45\cdot 53^{6} + 36\cdot 53^{7} + 11\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 9 + 10\cdot 53 + 42\cdot 53^{2} + 28\cdot 53^{3} + 37\cdot 53^{4} + 44\cdot 53^{5} + 7\cdot 53^{6} + 16\cdot 53^{7} + 41\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 27 a + 14 + \left(29 a + 47\right)\cdot 53 + 15 a\cdot 53^{2} + \left(26 a + 50\right)\cdot 53^{3} + \left(45 a + 21\right)\cdot 53^{4} + \left(24 a + 51\right)\cdot 53^{5} + \left(49 a + 49\right)\cdot 53^{6} + \left(51 a + 29\right)\cdot 53^{7} + \left(47 a + 12\right)\cdot 53^{8} +O\left(53^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,8,6)(3,4,7)$ |
| $(1,2,5,3)(4,7,8,6)$ |
| $(1,5)(2,3)(4,8)(6,7)$ |
| $(2,3)(4,6)(7,8)$ |
| $(1,6,5,7)(2,8,3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $12$ | $2$ | $(2,3)(4,6)(7,8)$ | $0$ |
| $8$ | $3$ | $(1,2,4)(3,8,5)$ | $-1$ |
| $6$ | $4$ | $(1,2,5,3)(4,7,8,6)$ | $0$ |
| $8$ | $6$ | $(1,8,2,5,4,3)(6,7)$ | $1$ |
| $6$ | $8$ | $(1,6,2,4,5,7,3,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $6$ | $8$ | $(1,7,2,8,5,6,3,4)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
