Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 34\cdot 59 + 19\cdot 59^{2} + 12\cdot 59^{3} + 32\cdot 59^{4} + 25\cdot 59^{5} + 55\cdot 59^{6} + 42\cdot 59^{7} + 4\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 a + 24 + \left(14 a + 57\right)\cdot 59 + \left(28 a + 51\right)\cdot 59^{2} + \left(38 a + 53\right)\cdot 59^{3} + \left(51 a + 22\right)\cdot 59^{4} + \left(36 a + 7\right)\cdot 59^{5} + \left(45 a + 25\right)\cdot 59^{6} + \left(29 a + 37\right)\cdot 59^{7} + \left(a + 43\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 a + 2 + \left(16 a + 19\right)\cdot 59 + \left(47 a + 14\right)\cdot 59^{2} + \left(52 a + 56\right)\cdot 59^{3} + \left(38 a + 6\right)\cdot 59^{4} + \left(28 a + 5\right)\cdot 59^{5} + \left(13 a + 37\right)\cdot 59^{6} + \left(35 a + 18\right)\cdot 59^{7} + \left(7 a + 43\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 24\cdot 59 + 39\cdot 59^{2} + 46\cdot 59^{3} + 26\cdot 59^{4} + 33\cdot 59^{5} + 3\cdot 59^{6} + 16\cdot 59^{7} + 54\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 a + 35 + \left(44 a + 1\right)\cdot 59 + \left(30 a + 7\right)\cdot 59^{2} + \left(20 a + 5\right)\cdot 59^{3} + \left(7 a + 36\right)\cdot 59^{4} + \left(22 a + 51\right)\cdot 59^{5} + \left(13 a + 33\right)\cdot 59^{6} + \left(29 a + 21\right)\cdot 59^{7} + \left(57 a + 15\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 57 + \left(42 a + 39\right)\cdot 59 + \left(11 a + 44\right)\cdot 59^{2} + \left(6 a + 2\right)\cdot 59^{3} + \left(20 a + 52\right)\cdot 59^{4} + \left(30 a + 53\right)\cdot 59^{5} + \left(45 a + 21\right)\cdot 59^{6} + \left(23 a + 40\right)\cdot 59^{7} + \left(51 a + 15\right)\cdot 59^{8} +O\left(59^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)$ |
| $(2,5)$ |
| $(1,3,2)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $4$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $4$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
| $4$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
