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:
| $r_{ 1 }$ |
$=$ |
$ 32 a + 15 + \left(29 a + 20\right)\cdot 59 + \left(9 a + 7\right)\cdot 59^{2} + \left(24 a + 31\right)\cdot 59^{3} + \left(5 a + 3\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 56 + 20\cdot 59 + 5\cdot 59^{2} + 41\cdot 59^{3} + 11\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 47 + \left(29 a + 17\right)\cdot 59 + \left(49 a + 46\right)\cdot 59^{2} + \left(34 a + 45\right)\cdot 59^{3} + \left(53 a + 43\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 a + 50 + \left(30 a + 30\right)\cdot 59 + \left(22 a + 23\right)\cdot 59^{2} + \left(18 a + 40\right)\cdot 59^{3} + \left(40 a + 43\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 + 48\cdot 59 + 19\cdot 59^{2} + 41\cdot 59^{3} + 8\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 36 a + 14 + \left(28 a + 38\right)\cdot 59 + \left(36 a + 15\right)\cdot 59^{2} + \left(40 a + 36\right)\cdot 59^{3} + \left(18 a + 6\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(4,5,6)$ |
| $(1,4)(2,5)(3,6)$ |
| $(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $2$ |
| $6$ | $2$ | $(1,2)$ | $0$ |
| $9$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $4$ | $3$ | $(1,2,3)$ | $-2$ |
| $18$ | $4$ | $(1,5,2,4)(3,6)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $-1$ |
| $12$ | $6$ | $(1,2)(4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
