Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{3} + 7 x + 59 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 a^{2} + 16 a + 46 + \left(16 a^{2} + 27 a + 1\right)\cdot 61 + \left(46 a^{2} + 25 a + 8\right)\cdot 61^{2} + \left(51 a^{2} + 4 a + 22\right)\cdot 61^{3} + \left(11 a^{2} + 4 a + 34\right)\cdot 61^{4} + \left(51 a^{2} + 34 a + 2\right)\cdot 61^{5} + \left(19 a^{2} + 21 a + 24\right)\cdot 61^{6} + \left(38 a^{2} + 21 a + 46\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 20\cdot 61 + 14\cdot 61^{2} + 54\cdot 61^{3} + 58\cdot 61^{4} + 58\cdot 61^{5} + 61^{6} + 40\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 a^{2} + 23 a + 41 + \left(51 a^{2} + 34 a + 5\right)\cdot 61 + \left(10 a^{2} + 22 a + 8\right)\cdot 61^{2} + \left(21 a^{2} + 28 a + 47\right)\cdot 61^{3} + \left(51 a^{2} + 25 a + 14\right)\cdot 61^{4} + \left(28 a^{2} + 9 a + 51\right)\cdot 61^{5} + \left(29 a^{2} + 38 a + 8\right)\cdot 61^{6} + \left(27 a^{2} + 38 a + 43\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 a^{2} + 49 a + 28 + \left(6 a^{2} + 10 a\right)\cdot 61 + \left(55 a^{2} + 48 a + 52\right)\cdot 61^{2} + \left(11 a^{2} + 41 a + 3\right)\cdot 61^{3} + \left(55 a^{2} + 24 a + 53\right)\cdot 61^{4} + \left(21 a^{2} + 18 a + 18\right)\cdot 61^{5} + \left(36 a^{2} + 60 a\right)\cdot 61^{6} + \left(30 a^{2} + 53 a + 17\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 a^{2} + 50 a + 41 + \left(3 a^{2} + 15 a + 25\right)\cdot 61 + \left(56 a^{2} + 51 a + 56\right)\cdot 61^{2} + \left(27 a^{2} + 51 a + 37\right)\cdot 61^{3} + \left(15 a^{2} + 10 a + 50\right)\cdot 61^{4} + \left(10 a^{2} + 33 a + 45\right)\cdot 61^{5} + \left(56 a^{2} + 23 a + 31\right)\cdot 61^{6} + \left(2 a^{2} + 29 a + 50\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a^{2} + 28 a + 16 + \left(37 a^{2} + 7 a + 19\right)\cdot 61 + \left(36 a^{2} + 37 a + 44\right)\cdot 61^{2} + \left(42 a^{2} + 29 a + 60\right)\cdot 61^{3} + \left(48 a^{2} + 2\right)\cdot 61^{4} + \left(17 a^{2} + 21 a + 50\right)\cdot 61^{5} + \left(29 a^{2} + 42 a + 47\right)\cdot 61^{6} + \left(60 a^{2} + 31 a + 27\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 47\cdot 61 + 3\cdot 61^{2} + 30\cdot 61^{3} + 9\cdot 61^{4} + 35\cdot 61^{5} + 20\cdot 61^{6} + 9\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 49 + 61 + 61^{2} + 58\cdot 61^{3} + 58\cdot 61^{4} + 10\cdot 61^{5} + 20\cdot 61^{6} + 54\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 6 a^{2} + 17 a + 12 + \left(7 a^{2} + 26 a + 60\right)\cdot 61 + \left(39 a^{2} + 59 a + 55\right)\cdot 61^{2} + \left(27 a^{2} + 26 a + 51\right)\cdot 61^{3} + \left(56 a + 21\right)\cdot 61^{4} + \left(53 a^{2} + 5 a + 31\right)\cdot 61^{5} + \left(11 a^{2} + 58 a + 27\right)\cdot 61^{6} + \left(23 a^{2} + 7 a + 16\right)\cdot 61^{7} +O\left(61^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(4,8,6)(5,7,9)$ |
| $(2,8,7)(3,5,4)$ |
| $(1,6,9)(2,7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $3$ | $(1,9,6)(2,7,8)(3,5,4)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,6,9)(2,8,7)(3,4,5)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(1,3,2)(4,8,6)(5,7,9)$ | $0$ |
| $3$ | $3$ | $(1,2,3)(4,6,8)(5,9,7)$ | $0$ |
| $3$ | $3$ | $(1,6,9)(2,7,8)$ | $0$ |
| $3$ | $3$ | $(1,9,6)(2,8,7)$ | $0$ |
| $3$ | $3$ | $(1,5,2)(3,8,6)(4,7,9)$ | $0$ |
| $3$ | $3$ | $(1,2,5)(3,6,8)(4,9,7)$ | $0$ |
| $3$ | $3$ | $(1,4,2)(3,7,9)(5,8,6)$ | $0$ |
| $3$ | $3$ | $(1,2,4)(3,9,7)(5,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
