Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 50\cdot 53 + 22\cdot 53^{2} + 49\cdot 53^{3} + 42\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 28 + \left(43 a + 17\right)\cdot 53 + \left(22 a + 45\right)\cdot 53^{2} + \left(11 a + 30\right)\cdot 53^{3} + \left(35 a + 13\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 a + 52 + \left(9 a + 24\right)\cdot 53 + \left(30 a + 40\right)\cdot 53^{2} + 41 a\cdot 53^{3} + \left(17 a + 37\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 18\cdot 53 + 50\cdot 53^{2} + 13\cdot 53^{3} + 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 47\cdot 53 + 52\cdot 53^{2} + 10\cdot 53^{3} + 11\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
