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:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 40 + \left(24 a + 2\right)\cdot 59 + \left(22 a + 30\right)\cdot 59^{2} + \left(18 a + 54\right)\cdot 59^{3} + \left(17 a + 18\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 a + 9 + \left(26 a + 55\right)\cdot 59 + \left(58 a + 57\right)\cdot 59^{2} + \left(21 a + 43\right)\cdot 59^{3} + \left(33 a + 15\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 10 + \left(8 a + 1\right)\cdot 59 + \left(37 a + 30\right)\cdot 59^{2} + \left(18 a + 19\right)\cdot 59^{3} + \left(8 a + 24\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 48 a + 21 + \left(50 a + 57\right)\cdot 59 + \left(21 a + 58\right)\cdot 59^{2} + 40 a\cdot 59^{3} + \left(50 a + 14\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 a + 43 + \left(34 a + 23\right)\cdot 59 + \left(36 a + 28\right)\cdot 59^{2} + \left(40 a + 50\right)\cdot 59^{3} + \left(41 a + 17\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 a + 54 + \left(32 a + 36\right)\cdot 59 + 30\cdot 59^{2} + \left(37 a + 7\right)\cdot 59^{3} + \left(25 a + 27\right)\cdot 59^{4} +O\left(59^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,3,5,2,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,2,5,3,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
