Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: $ x^{2} + 58 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 + 49\cdot 59 + 13\cdot 59^{2} + 5\cdot 59^{3} + 28\cdot 59^{4} + 54\cdot 59^{5} + 16\cdot 59^{6} + 26\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 55 + \left(52 a + 36\right)\cdot 59 + \left(34 a + 8\right)\cdot 59^{2} + \left(4 a + 15\right)\cdot 59^{3} + \left(46 a + 38\right)\cdot 59^{4} + \left(11 a + 46\right)\cdot 59^{5} + \left(30 a + 49\right)\cdot 59^{6} + \left(45 a + 21\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 a + 24 + \left(8 a + 33\right)\cdot 59 + \left(28 a + 17\right)\cdot 59^{2} + \left(40 a + 44\right)\cdot 59^{3} + \left(38 a + 26\right)\cdot 59^{4} + \left(29 a + 48\right)\cdot 59^{5} + \left(a + 53\right)\cdot 59^{6} + \left(7 a + 42\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 27 + \left(8 a + 25\right)\cdot 59 + \left(28 a + 21\right)\cdot 59^{2} + \left(40 a + 2\right)\cdot 59^{3} + \left(38 a + 34\right)\cdot 59^{4} + \left(29 a + 19\right)\cdot 59^{5} + \left(a + 33\right)\cdot 59^{6} + \left(7 a + 10\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 9\cdot 59 + 45\cdot 59^{2} + 53\cdot 59^{3} + 30\cdot 59^{4} + 4\cdot 59^{5} + 42\cdot 59^{6} + 32\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 a + 4 + \left(6 a + 22\right)\cdot 59 + \left(24 a + 50\right)\cdot 59^{2} + \left(54 a + 43\right)\cdot 59^{3} + \left(12 a + 20\right)\cdot 59^{4} + \left(47 a + 12\right)\cdot 59^{5} + \left(28 a + 9\right)\cdot 59^{6} + \left(13 a + 37\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 51 a + 35 + \left(50 a + 25\right)\cdot 59 + \left(30 a + 41\right)\cdot 59^{2} + \left(18 a + 14\right)\cdot 59^{3} + \left(20 a + 32\right)\cdot 59^{4} + \left(29 a + 10\right)\cdot 59^{5} + \left(57 a + 5\right)\cdot 59^{6} + \left(51 a + 16\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 51 a + 32 + \left(50 a + 33\right)\cdot 59 + \left(30 a + 37\right)\cdot 59^{2} + \left(18 a + 56\right)\cdot 59^{3} + \left(20 a + 24\right)\cdot 59^{4} + \left(29 a + 39\right)\cdot 59^{5} + \left(57 a + 25\right)\cdot 59^{6} + \left(51 a + 48\right)\cdot 59^{7} +O\left(59^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,5,4)(2,7,6,3)$ |
| $(1,5)(2,8)(4,6)$ |
| $(1,6,8)(2,4,5)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
| $(1,7,5,3)(2,4,6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(1,5)(2,8)(4,6)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,6,8)(2,4,5)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,7,5,3)(2,4,6,8)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,8,7,5,4,3)(2,6)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,4,3,2,5,8,7,6)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,8,3,6,5,4,7,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
