Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 35 + \left(15 a + 11\right)\cdot 53 + \left(38 a + 41\right)\cdot 53^{2} + \left(29 a + 40\right)\cdot 53^{3} + \left(36 a + 4\right)\cdot 53^{4} + \left(20 a + 15\right)\cdot 53^{5} + \left(27 a + 2\right)\cdot 53^{6} + \left(35 a + 44\right)\cdot 53^{7} + \left(25 a + 7\right)\cdot 53^{8} + \left(51 a + 43\right)\cdot 53^{9} +O\left(53^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 49 + \left(48 a + 2\right)\cdot 53 + \left(4 a + 41\right)\cdot 53^{2} + \left(50 a + 34\right)\cdot 53^{3} + \left(15 a + 19\right)\cdot 53^{4} + \left(34 a + 45\right)\cdot 53^{5} + \left(42 a + 37\right)\cdot 53^{6} + \left(25 a + 22\right)\cdot 53^{7} + \left(14 a + 10\right)\cdot 53^{8} + \left(a + 31\right)\cdot 53^{9} +O\left(53^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 16\cdot 53 + 45\cdot 53^{2} + 8\cdot 53^{3} + 38\cdot 53^{4} + 41\cdot 53^{5} + 22\cdot 53^{6} + 12\cdot 53^{7} + 20\cdot 53^{8} + 48\cdot 53^{9} +O\left(53^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 36\cdot 53 + 7\cdot 53^{2} + 44\cdot 53^{3} + 14\cdot 53^{4} + 11\cdot 53^{5} + 30\cdot 53^{6} + 40\cdot 53^{7} + 32\cdot 53^{8} + 4\cdot 53^{9} +O\left(53^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 3 a + 7 + \left(15 a + 37\right)\cdot 53 + \left(38 a + 32\right)\cdot 53^{2} + \left(29 a + 37\right)\cdot 53^{3} + \left(36 a + 37\right)\cdot 53^{4} + \left(20 a + 44\right)\cdot 53^{5} + \left(27 a + 14\right)\cdot 53^{6} + 35 a\cdot 53^{7} + \left(25 a + 31\right)\cdot 53^{8} + \left(51 a + 41\right)\cdot 53^{9} +O\left(53^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 5 + \left(4 a + 50\right)\cdot 53 + \left(48 a + 11\right)\cdot 53^{2} + \left(2 a + 18\right)\cdot 53^{3} + \left(37 a + 33\right)\cdot 53^{4} + \left(18 a + 7\right)\cdot 53^{5} + \left(10 a + 15\right)\cdot 53^{6} + \left(27 a + 30\right)\cdot 53^{7} + \left(38 a + 42\right)\cdot 53^{8} + \left(51 a + 21\right)\cdot 53^{9} +O\left(53^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 50 a + 19 + \left(37 a + 41\right)\cdot 53 + \left(14 a + 11\right)\cdot 53^{2} + \left(23 a + 12\right)\cdot 53^{3} + \left(16 a + 48\right)\cdot 53^{4} + \left(32 a + 37\right)\cdot 53^{5} + \left(25 a + 50\right)\cdot 53^{6} + \left(17 a + 8\right)\cdot 53^{7} + \left(27 a + 45\right)\cdot 53^{8} + \left(a + 9\right)\cdot 53^{9} +O\left(53^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 50 a + 47 + \left(37 a + 15\right)\cdot 53 + \left(14 a + 20\right)\cdot 53^{2} + \left(23 a + 15\right)\cdot 53^{3} + \left(16 a + 15\right)\cdot 53^{4} + \left(32 a + 8\right)\cdot 53^{5} + \left(25 a + 38\right)\cdot 53^{6} + \left(17 a + 52\right)\cdot 53^{7} + \left(27 a + 21\right)\cdot 53^{8} + \left(a + 11\right)\cdot 53^{9} +O\left(53^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6)(5,7)$ |
| $(1,6,5)(2,8,7)$ |
| $(1,3,7,4)(2,5,6,8)$ |
| $(1,6,7,2)(3,5,4,8)$ |
| $(1,7)(2,6)(3,4)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,6)(3,4)(5,8)$ | $-4$ |
| $12$ | $2$ | $(1,8)(2,6)(5,7)$ | $0$ |
| $8$ | $3$ | $(1,3,2)(4,6,7)$ | $1$ |
| $6$ | $4$ | $(1,3,7,4)(2,5,6,8)$ | $0$ |
| $8$ | $6$ | $(1,7)(2,3,5,6,4,8)$ | $-1$ |
| $6$ | $8$ | $(1,2,8,3,7,6,5,4)$ | $0$ |
| $6$ | $8$ | $(1,6,8,4,7,2,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
