Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 a + 16 + \left(42 a + 6\right)\cdot 59 + \left(13 a + 19\right)\cdot 59^{2} + \left(5 a + 11\right)\cdot 59^{3} + \left(40 a + 47\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 + 49\cdot 59 + 25\cdot 59^{2} + 3\cdot 59^{3} + 24\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 57 + 4\cdot 59 + 29\cdot 59^{2} + 52\cdot 59^{3} + 32\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a + \left(25 a + 44\right)\cdot 59 + \left(30 a + 24\right)\cdot 59^{2} + \left(31 a + 23\right)\cdot 59^{3} + \left(4 a + 38\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 57 + \left(33 a + 12\right)\cdot 59 + \left(28 a + 29\right)\cdot 59^{2} + \left(27 a + 24\right)\cdot 59^{3} + \left(54 a + 11\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 a + 6 + 16 a\cdot 59 + \left(45 a + 49\right)\cdot 59^{2} + \left(53 a + 2\right)\cdot 59^{3} + \left(18 a + 23\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
| $(1,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(1,3)$ | $0$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $1$ |
| $18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,4,3,5,6,2)$ | $-1$ |
| $12$ | $6$ | $(1,3)(2,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
