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:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 49 + \left(39 a + 21\right)\cdot 59 + \left(30 a + 47\right)\cdot 59^{2} + \left(54 a + 8\right)\cdot 59^{3} + \left(7 a + 26\right)\cdot 59^{4} + \left(16 a + 32\right)\cdot 59^{5} + \left(44 a + 43\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 54 a + 54 + \left(19 a + 55\right)\cdot 59 + \left(28 a + 38\right)\cdot 59^{2} + \left(4 a + 32\right)\cdot 59^{3} + \left(51 a + 38\right)\cdot 59^{4} + \left(42 a + 40\right)\cdot 59^{5} + \left(14 a + 12\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 47\cdot 59 + 26\cdot 59^{2} + 2\cdot 59^{3} + 39\cdot 59^{4} + 58\cdot 59^{5} + 42\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 50\cdot 59 + 12\cdot 59^{2} + 34\cdot 59^{3} + 21\cdot 59^{4} + 30\cdot 59^{5} + 22\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 18 + \left(35 a + 28\right)\cdot 59 + \left(15 a + 35\right)\cdot 59^{2} + \left(37 a + 38\right)\cdot 59^{3} + \left(41 a + 23\right)\cdot 59^{4} + \left(27 a + 14\right)\cdot 59^{5} + 23 a\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a + 50 + \left(23 a + 31\right)\cdot 59 + \left(43 a + 15\right)\cdot 59^{2} + \left(21 a + 1\right)\cdot 59^{3} + \left(17 a + 28\right)\cdot 59^{4} + 31 a\cdot 59^{5} + \left(35 a + 55\right)\cdot 59^{6} +O\left(59^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)$ |
| $(1,5,3)(2,6,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,6)(3,4)$ | $1$ |
| $8$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
| $6$ | $4$ | $(1,3,2,4)$ | $-1$ |
| $6$ | $4$ | $(1,2)(3,6,4,5)$ | $1$ |
| $8$ | $6$ | $(1,5,3,2,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
