Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 35 a + 41 + \left(45 a + 5\right)\cdot 59 + \left(44 a + 49\right)\cdot 59^{2} + \left(13 a + 13\right)\cdot 59^{3} + \left(11 a + 30\right)\cdot 59^{4} + \left(37 a + 53\right)\cdot 59^{5} + \left(20 a + 19\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 a + 30 + \left(45 a + 40\right)\cdot 59 + \left(44 a + 29\right)\cdot 59^{2} + \left(13 a + 15\right)\cdot 59^{3} + \left(11 a + 32\right)\cdot 59^{4} + \left(37 a + 40\right)\cdot 59^{5} + \left(20 a + 12\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 a + 6 + \left(13 a + 51\right)\cdot 59 + \left(14 a + 28\right)\cdot 59^{2} + \left(45 a + 43\right)\cdot 59^{3} + \left(47 a + 29\right)\cdot 59^{4} + \left(21 a + 7\right)\cdot 59^{5} + \left(38 a + 55\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 + 43\cdot 59 + 49\cdot 59^{2} + 57\cdot 59^{4} + 33\cdot 59^{5} + 46\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 19\cdot 59 + 30\cdot 59^{2} + 2\cdot 59^{3} + 21\cdot 59^{5} + 39\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 a + 17 + \left(13 a + 16\right)\cdot 59 + \left(14 a + 48\right)\cdot 59^{2} + \left(45 a + 41\right)\cdot 59^{3} + \left(47 a + 27\right)\cdot 59^{4} + \left(21 a + 20\right)\cdot 59^{5} + \left(38 a + 3\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(3,5)(4,6)$ |
| $(1,3,4,2,6,5)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,6)(2,5,3)$ | $-1$ |
| $2$ | $6$ | $(1,3,4,2,6,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
