Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 a + 29 + \left(28 a + 37\right)\cdot 59 + \left(25 a + 58\right)\cdot 59^{2} + \left(50 a + 47\right)\cdot 59^{3} + \left(a + 50\right)\cdot 59^{4} + \left(42 a + 16\right)\cdot 59^{5} + \left(9 a + 43\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 + 16\cdot 59 + 12\cdot 59^{2} + 33\cdot 59^{3} + 55\cdot 59^{4} + 15\cdot 59^{5} + 31\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 43 + \left(58 a + 10\right)\cdot 59 + \left(16 a + 49\right)\cdot 59^{2} + \left(3 a + 39\right)\cdot 59^{3} + \left(37 a + 29\right)\cdot 59^{4} + \left(46 a + 18\right)\cdot 59^{5} + \left(46 a + 32\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 31 a + 57 + \left(30 a + 37\right)\cdot 59 + \left(33 a + 55\right)\cdot 59^{2} + \left(8 a + 13\right)\cdot 59^{3} + \left(57 a + 2\right)\cdot 59^{4} + \left(16 a + 57\right)\cdot 59^{5} + \left(49 a + 10\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 16\cdot 59 + 52\cdot 59^{2} + 15\cdot 59^{3} + 34\cdot 59^{4} + 40\cdot 59^{5} + 26\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 a + 54 + 57\cdot 59 + \left(42 a + 7\right)\cdot 59^{2} + \left(55 a + 26\right)\cdot 59^{3} + \left(21 a + 4\right)\cdot 59^{4} + \left(12 a + 28\right)\cdot 59^{5} + \left(12 a + 32\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)(2,6,4)$ |
| $(1,5)(2,6)$ |
| $(1,2)(5,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,6)(2,5)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,5)(2,6,4)$ | $0$ |
| $6$ | $4$ | $(1,4,6,3)(2,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
