Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 13\cdot 47 + 2\cdot 47^{2} + 6\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 a + \left(39 a + 20\right)\cdot 47 + \left(43 a + 40\right)\cdot 47^{2} + \left(12 a + 1\right)\cdot 47^{3} + 15\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 a + 26 + \left(7 a + 38\right)\cdot 47 + \left(3 a + 41\right)\cdot 47^{2} + \left(34 a + 30\right)\cdot 47^{3} + \left(46 a + 2\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 a + 29 + \left(25 a + 36\right)\cdot 47 + 35 a\cdot 47^{2} + \left(18 a + 25\right)\cdot 47^{3} + \left(4 a + 8\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 + 33\cdot 47 + 9\cdot 47^{2} + 3\cdot 47^{3} + 43\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a + 19 + \left(21 a + 46\right)\cdot 47 + \left(11 a + 45\right)\cdot 47^{2} + \left(28 a + 26\right)\cdot 47^{3} + \left(42 a + 45\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6,4,3,2,5)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,4)(2,3)(5,6)$ | $1$ |
| $15$ | $2$ | $(1,5)(2,3)$ | $1$ |
| $20$ | $3$ | $(1,4,2)(3,5,6)$ | $-1$ |
| $30$ | $4$ | $(1,2,5,3)$ | $-1$ |
| $24$ | $5$ | $(1,6,2,4,5)$ | $0$ |
| $20$ | $6$ | $(1,6,4,3,2,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
