Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 a + 19 + 33\cdot 59 + \left(43 a + 57\right)\cdot 59^{2} + \left(58 a + 39\right)\cdot 59^{3} + \left(8 a + 25\right)\cdot 59^{4} + \left(30 a + 4\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 47 + \left(58 a + 5\right)\cdot 59 + \left(15 a + 41\right)\cdot 59^{2} + 55\cdot 59^{3} + \left(50 a + 34\right)\cdot 59^{4} + \left(28 a + 25\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a + 32 + \left(17 a + 49\right)\cdot 59 + \left(7 a + 18\right)\cdot 59^{2} + \left(9 a + 49\right)\cdot 59^{3} + \left(56 a + 8\right)\cdot 59^{4} + \left(39 a + 15\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 + 20\cdot 59^{2} + 4\cdot 59^{3} + 39\cdot 59^{4} + 51\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 43 + 31\cdot 59^{2} + 35\cdot 59^{3} + 12\cdot 59^{4} + 22\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 a + 12 + \left(41 a + 28\right)\cdot 59 + \left(51 a + 8\right)\cdot 59^{2} + \left(49 a + 51\right)\cdot 59^{3} + \left(2 a + 55\right)\cdot 59^{4} + \left(19 a + 57\right)\cdot 59^{5} +O\left(59^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,6,5)$ |
| $(3,4)(5,6)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,2)$ |
$1$ |
| $3$ |
$2$ |
$(1,2)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(3,4)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,4)(2,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,6,2,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,5,2,4)(3,6)$ |
$1$ |
| $8$ |
$6$ |
$(1,6,5,2,3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
