Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{3} + 7 x + 59 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 23\cdot 61 + 7\cdot 61^{2} + 31\cdot 61^{3} + 61^{4} + 59\cdot 61^{5} + 58\cdot 61^{6} + 51\cdot 61^{7} + 42\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 a^{2} + 19 a + 60 + \left(15 a^{2} + 48 a + 38\right)\cdot 61 + \left(40 a^{2} + 12 a + 16\right)\cdot 61^{2} + \left(10 a^{2} + 22 a + 52\right)\cdot 61^{3} + \left(27 a^{2} + 27 a + 5\right)\cdot 61^{4} + \left(44 a^{2} + 12 a + 53\right)\cdot 61^{5} + \left(33 a^{2} + 37 a + 35\right)\cdot 61^{6} + \left(14 a^{2} + 43 a + 17\right)\cdot 61^{7} + \left(19 a^{2} + 36 a + 3\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 7\cdot 61 + 30\cdot 61^{2} + 47\cdot 61^{3} + 40\cdot 61^{4} + 47\cdot 61^{5} + 4\cdot 61^{6} + 25\cdot 61^{7} + 53\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a^{2} + 4 a + 59 + \left(43 a^{2} + 29 a + 45\right)\cdot 61 + \left(30 a^{2} + 18 a + 12\right)\cdot 61^{2} + \left(24 a^{2} + 24 a + 15\right)\cdot 61^{3} + \left(20 a^{2} + 44 a + 35\right)\cdot 61^{4} + \left(20 a^{2} + 33 a + 1\right)\cdot 61^{5} + \left(57 a^{2} + 33 a + 24\right)\cdot 61^{6} + \left(25 a^{2} + 45 a + 50\right)\cdot 61^{7} + \left(25 a^{2} + 50 a + 11\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a^{2} + 51 a + 32 + \left(60 a^{2} + 21 a + 33\right)\cdot 61 + \left(6 a^{2} + 19 a + 40\right)\cdot 61^{2} + \left(6 a^{2} + 8 a + 14\right)\cdot 61^{3} + \left(49 a^{2} + a + 23\right)\cdot 61^{4} + \left(11 a^{2} + 45 a + 56\right)\cdot 61^{5} + \left(12 a^{2} + 19 a + 36\right)\cdot 61^{6} + \left(59 a^{2} + 25 a + 2\right)\cdot 61^{7} + \left(15 a^{2} + 45 a + 22\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ a^{2} + 38 a + 36 + \left(2 a^{2} + 44 a + 56\right)\cdot 61 + \left(51 a^{2} + 29 a + 5\right)\cdot 61^{2} + \left(25 a^{2} + 14 a + 1\right)\cdot 61^{3} + \left(13 a^{2} + 50 a + 23\right)\cdot 61^{4} + \left(57 a^{2} + 14 a + 31\right)\cdot 61^{5} + \left(30 a^{2} + 51 a + 2\right)\cdot 61^{6} + \left(20 a^{2} + 32 a + 25\right)\cdot 61^{7} + \left(16 a^{2} + 34 a + 30\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 a^{2} + 12 a + 18 + \left(56 a^{2} + 49 a + 55\right)\cdot 61 + \left(49 a^{2} + 55 a + 57\right)\cdot 61^{2} + \left(4 a^{2} + 32 a + 28\right)\cdot 61^{3} + \left(45 a^{2} + 6 a + 4\right)\cdot 61^{4} + \left(20 a^{2} + 45 a + 37\right)\cdot 61^{5} + \left(12 a^{2} + 19 a + 37\right)\cdot 61^{6} + \left(28 a^{2} + 41 a + 20\right)\cdot 61^{7} + \left(a^{2} + 17 a + 15\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 49 a^{2} + 59 a + 2 + \left(4 a^{2} + 50 a + 17\right)\cdot 61 + \left(4 a^{2} + 46 a + 47\right)\cdot 61^{2} + \left(50 a^{2} + 19 a + 36\right)\cdot 61^{3} + \left(27 a^{2} + 53 a + 25\right)\cdot 61^{4} + \left(28 a^{2} + 31 a + 32\right)\cdot 61^{5} + \left(36 a^{2} + 21 a + 48\right)\cdot 61^{6} + \left(34 a^{2} + 55 a + 9\right)\cdot 61^{7} + \left(43 a^{2} + 58 a + 49\right)\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 28 + 27\cdot 61 + 25\cdot 61^{2} + 16\cdot 61^{3} + 23\cdot 61^{4} + 47\cdot 61^{5} + 55\cdot 61^{6} + 40\cdot 61^{7} + 15\cdot 61^{8} +O\left(61^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,6,4)(5,8,7)$ |
| $(1,9,3)(2,6,4)$ |
| $(1,7,6,3,8,4,9,5,2)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)(3,4)(6,9)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,3,9)(2,6,4)(5,7,8)$ | $-3$ |
| $3$ | $3$ | $(1,9,3)(2,6,4)$ | $0$ |
| $3$ | $3$ | $(1,3,9)(2,4,6)$ | $0$ |
| $9$ | $6$ | $(1,4,9,2,3,6)(7,8)$ | $0$ |
| $9$ | $6$ | $(1,6,3,2,9,4)(7,8)$ | $0$ |
| $6$ | $9$ | $(1,7,6,3,8,4,9,5,2)$ | $0$ |
| $6$ | $9$ | $(1,6,8,9,2,7,3,4,5)$ | $0$ |
| $6$ | $9$ | $(1,8,6,3,5,4,9,7,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
