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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 8 + \left(36 a + 41\right)\cdot 59 + \left(25 a + 31\right)\cdot 59^{2} + \left(9 a + 58\right)\cdot 59^{3} + \left(52 a + 1\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 a + 28 + \left(22 a + 57\right)\cdot 59 + \left(33 a + 20\right)\cdot 59^{2} + \left(49 a + 42\right)\cdot 59^{3} + \left(6 a + 44\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 13 + \left(9 a + 36\right)\cdot 59 + \left(6 a + 55\right)\cdot 59^{2} + 9 a\cdot 59^{3} + \left(15 a + 13\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 51\cdot 59 + 9\cdot 59^{2} + 54\cdot 59^{3} + 26\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 a + 28 + \left(49 a + 30\right)\cdot 59 + \left(52 a + 52\right)\cdot 59^{2} + \left(49 a + 3\right)\cdot 59^{3} + \left(43 a + 19\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 + 19\cdot 59 + 6\cdot 59^{2} + 17\cdot 59^{3} + 12\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)$ |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$2$ |
| $6$ |
$2$ |
$(4,5)$ |
$0$ |
| $9$ |
$2$ |
$(2,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$1$ |
| $4$ |
$3$ |
$(3,4,5)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,4,6,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,2,4,6,5)$ |
$-1$ |
| $12$ |
$6$ |
$(1,2,6)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
