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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 4 + \left(39 a + 13\right)\cdot 53 + \left(7 a + 13\right)\cdot 53^{2} + \left(34 a + 34\right)\cdot 53^{3} + \left(45 a + 32\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 22 + \left(13 a + 34\right)\cdot 53 + \left(45 a + 4\right)\cdot 53^{2} + \left(18 a + 4\right)\cdot 53^{3} + \left(7 a + 22\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 a + 46 + \left(37 a + 43\right)\cdot 53 + \left(50 a + 38\right)\cdot 53^{2} + \left(27 a + 45\right)\cdot 53^{3} + \left(24 a + 15\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 28\cdot 53 + 4\cdot 53^{2} + 21\cdot 53^{3} + 2\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 38 + \left(15 a + 38\right)\cdot 53 + \left(2 a + 44\right)\cdot 53^{2} + 25 a\cdot 53^{3} + \left(28 a + 33\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
