Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 35 + \left(25 a + 52\right)\cdot 53 + \left(26 a + 4\right)\cdot 53^{2} + \left(48 a + 9\right)\cdot 53^{3} + \left(17 a + 21\right)\cdot 53^{4} + \left(52 a + 5\right)\cdot 53^{5} + \left(17 a + 39\right)\cdot 53^{6} + \left(37 a + 1\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 a + 9 + \left(26 a + 16\right)\cdot 53 + \left(12 a + 16\right)\cdot 53^{2} + \left(3 a + 35\right)\cdot 53^{3} + \left(17 a + 27\right)\cdot 53^{4} + \left(29 a + 23\right)\cdot 53^{5} + \left(5 a + 20\right)\cdot 53^{6} + \left(48 a + 12\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 39 + \left(26 a + 35\right)\cdot 53 + \left(40 a + 39\right)\cdot 53^{2} + \left(49 a + 35\right)\cdot 53^{3} + \left(35 a + 39\right)\cdot 53^{4} + \left(23 a + 17\right)\cdot 53^{5} + \left(47 a + 13\right)\cdot 53^{6} + \left(4 a + 40\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 a + 25 + \left(27 a + 24\right)\cdot 53 + \left(26 a + 32\right)\cdot 53^{2} + \left(4 a + 17\right)\cdot 53^{3} + \left(35 a + 44\right)\cdot 53^{4} + 37\cdot 53^{5} + \left(35 a + 5\right)\cdot 53^{6} + \left(15 a + 27\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 32 + \left(a + 17\right)\cdot 53 + \left(39 a + 8\right)\cdot 53^{2} + \left(7 a + 8\right)\cdot 53^{3} + \left(52 a + 45\right)\cdot 53^{4} + \left(29 a + 29\right)\cdot 53^{5} + 40 a\cdot 53^{6} + \left(10 a + 11\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 43 a + 19 + \left(51 a + 12\right)\cdot 53 + \left(13 a + 4\right)\cdot 53^{2} + 45 a\cdot 53^{3} + 34\cdot 53^{4} + \left(23 a + 44\right)\cdot 53^{5} + \left(12 a + 26\right)\cdot 53^{6} + \left(42 a + 13\right)\cdot 53^{7} +O\left(53^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)$ |
| $(1,4,3,2,5,6)$ |
| $(1,3,5)(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $-2 \zeta_{3} - 2$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $2 \zeta_{3}$ |
| $2$ | $3$ | $(1,3,5)(2,4,6)$ | $-1$ |
| $2$ | $3$ | $(1,3,5)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(1,5,3)$ | $\zeta_{3} + 1$ |
| $3$ | $6$ | $(1,4,3,2,5,6)$ | $0$ |
| $3$ | $6$ | $(1,6,5,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
