Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{3} + 7 x + 59 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 36 + 27\cdot 61 + 3\cdot 61^{2} + 38\cdot 61^{3} + 38\cdot 61^{4} + 29\cdot 61^{5} + 36\cdot 61^{6} + 24\cdot 61^{7} + 23\cdot 61^{8} + 42\cdot 61^{9} + 16\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 44 a^{2} + 3 a + 2 + \left(48 a^{2} + 14 a + 24\right)\cdot 61 + \left(23 a^{2} + 7 a + 9\right)\cdot 61^{2} + \left(14 a^{2} + 42 a + 6\right)\cdot 61^{3} + \left(49 a^{2} + 46 a + 6\right)\cdot 61^{4} + \left(10 a^{2} + 49 a + 30\right)\cdot 61^{5} + \left(42 a^{2} + 33 a + 54\right)\cdot 61^{6} + \left(13 a^{2} + 48 a + 2\right)\cdot 61^{7} + \left(58 a^{2} + 38 a + 48\right)\cdot 61^{8} + \left(55 a^{2} + 42 a + 57\right)\cdot 61^{9} + \left(49 a^{2} + 34 a + 49\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a^{2} + 13 a + 1 + \left(23 a^{2} + 6 a + 48\right)\cdot 61 + \left(37 a^{2} + 5 a + 11\right)\cdot 61^{2} + \left(45 a^{2} + 56 a + 50\right)\cdot 61^{3} + \left(41 a^{2} + 36 a + 11\right)\cdot 61^{4} + \left(28 a^{2} + 30 a + 32\right)\cdot 61^{5} + \left(3 a^{2} + 18 a + 36\right)\cdot 61^{6} + \left(60 a^{2} + 58 a + 56\right)\cdot 61^{7} + \left(10 a^{2} + 35 a + 30\right)\cdot 61^{8} + \left(17 a^{2} + 18 a + 39\right)\cdot 61^{9} + \left(12 a^{2} + 58 a + 16\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 44\cdot 61 + 11\cdot 61^{2} + 46\cdot 61^{3} + 53\cdot 61^{4} + 7\cdot 61^{5} + 53\cdot 61^{6} + 21\cdot 61^{7} + 29\cdot 61^{8} + 49\cdot 61^{9} + 32\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 a^{2} + 19 a + 33 + \left(20 a^{2} + 27 a + 36\right)\cdot 61 + \left(33 a^{2} + 8 a + 33\right)\cdot 61^{2} + \left(43 a^{2} + 12 a + 40\right)\cdot 61^{3} + \left(26 a^{2} + 23 a + 2\right)\cdot 61^{4} + \left(36 a^{2} + a + 48\right)\cdot 61^{5} + \left(56 a^{2} + 47 a + 60\right)\cdot 61^{6} + \left(48 a^{2} + 51 a + 24\right)\cdot 61^{7} + \left(42 a^{2} + 50 a + 57\right)\cdot 61^{8} + \left(20 a^{2} + 54 a + 55\right)\cdot 61^{9} + \left(26 a^{2} + 33 a\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a^{2} + a + 47 + \left(54 a^{2} + 50 a + 32\right)\cdot 61 + \left(52 a^{2} + 49 a + 43\right)\cdot 61^{2} + \left(35 a^{2} + 4 a + 4\right)\cdot 61^{3} + \left(22 a^{2} + 2 a + 24\right)\cdot 61^{4} + \left(3 a^{2} + 19 a + 56\right)\cdot 61^{5} + \left(24 a^{2} + 30\right)\cdot 61^{6} + \left(21 a^{2} + 27 a + 18\right)\cdot 61^{7} + \left(43 a^{2} + 7 a + 19\right)\cdot 61^{8} + \left(49 a^{2} + 51 a + 8\right)\cdot 61^{9} + \left(29 a^{2} + 49 a + 17\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 + 49\cdot 61 + 45\cdot 61^{2} + 37\cdot 61^{3} + 29\cdot 61^{4} + 23\cdot 61^{5} + 32\cdot 61^{6} + 14\cdot 61^{7} + 8\cdot 61^{8} + 30\cdot 61^{9} + 11\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 23 a^{2} + 39 a + 26 + \left(52 a^{2} + 19 a\right)\cdot 61 + \left(3 a^{2} + 45 a + 18\right)\cdot 61^{2} + \left(3 a^{2} + 6 a + 14\right)\cdot 61^{3} + \left(46 a^{2} + 52 a + 52\right)\cdot 61^{4} + \left(13 a^{2} + 9 a + 43\right)\cdot 61^{5} + \left(23 a^{2} + 41 a + 6\right)\cdot 61^{6} + \left(59 a^{2} + 21 a + 33\right)\cdot 61^{7} + \left(20 a^{2} + 32 a + 16\right)\cdot 61^{8} + \left(45 a^{2} + 24 a + 8\right)\cdot 61^{9} + \left(45 a^{2} + 53 a + 10\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 42 a^{2} + 47 a + 13 + \left(43 a^{2} + 4 a + 41\right)\cdot 61 + \left(31 a^{2} + 6 a + 5\right)\cdot 61^{2} + \left(40 a^{2} + 6\right)\cdot 61^{3} + \left(57 a^{2} + 22 a + 25\right)\cdot 61^{4} + \left(28 a^{2} + 11 a + 33\right)\cdot 61^{5} + \left(33 a^{2} + 42 a + 54\right)\cdot 61^{6} + \left(40 a^{2} + 36 a + 46\right)\cdot 61^{7} + \left(6 a^{2} + 17 a + 10\right)\cdot 61^{8} + \left(55 a^{2} + 52 a + 13\right)\cdot 61^{9} + \left(18 a^{2} + 13 a + 27\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,6)(3,8)(5,9)$ |
| $(1,3,8)(2,4,6)(5,7,9)$ |
| $(2,5)(4,7)(6,9)$ |
| $(1,4,7)(2,5,8)(3,6,9)$ |
| $(2,5,8)(3,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,6)(3,8)(5,9)$ | $-2$ |
| $9$ | $2$ | $(2,5)(4,7)(6,9)$ | $0$ |
| $9$ | $2$ | $(2,9)(3,8)(4,7)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,4,7)(2,5,8)(3,6,9)$ | $-3$ |
| $6$ | $3$ | $(1,3,8)(2,4,6)(5,7,9)$ | $0$ |
| $6$ | $3$ | $(2,5,8)(3,9,6)$ | $0$ |
| $12$ | $3$ | $(1,9,8)(2,4,3)(5,7,6)$ | $0$ |
| $18$ | $6$ | $(1,4,7)(2,9,8,6,5,3)$ | $1$ |
| $18$ | $6$ | $(1,3,8)(2,7,6,5,4,9)$ | $0$ |
| $18$ | $6$ | $(1,9,4,6,7,3)(2,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
