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:
| $r_{ 1 }$ |
$=$ |
$ 50 a^{2} + 21 a + 17 + \left(31 a^{2} + 8 a + 9\right)\cdot 61 + \left(50 a^{2} + 6 a + 40\right)\cdot 61^{2} + \left(31 a^{2} + 39 a + 55\right)\cdot 61^{3} + \left(36 a^{2} + 50 a + 40\right)\cdot 61^{4} + \left(59 a^{2} + 30 a + 46\right)\cdot 61^{5} + \left(23 a^{2} + 45 a + 44\right)\cdot 61^{6} + \left(55 a^{2} + 46 a + 32\right)\cdot 61^{7} + \left(9 a^{2} + 54 a + 57\right)\cdot 61^{8} + \left(8 a^{2} + 3 a + 38\right)\cdot 61^{9} + \left(15 a^{2} + 27 a + 43\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 a^{2} + 45 a + 35 + \left(21 a^{2} + 51 a + 46\right)\cdot 61 + \left(32 a^{2} + 60 a + 39\right)\cdot 61^{2} + \left(3 a^{2} + 30 a + 45\right)\cdot 61^{3} + \left(52 a^{2} + 13 a + 33\right)\cdot 61^{4} + \left(58 a^{2} + 50 a + 43\right)\cdot 61^{5} + \left(26 a^{2} + 19 a + 8\right)\cdot 61^{6} + \left(a^{2} + 4 a + 45\right)\cdot 61^{7} + \left(40 a + 53\right)\cdot 61^{8} + \left(27 a^{2} + 10 a + 56\right)\cdot 61^{9} + \left(58 a^{2} + 43 a + 15\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a^{2} + 15 a + 37 + \left(44 a^{2} + 44 a + 30\right)\cdot 61 + \left(25 a^{2} + 27 a + 49\right)\cdot 61^{2} + \left(22 a^{2} + 28 a + 11\right)\cdot 61^{3} + \left(32 a^{2} + 2 a + 23\right)\cdot 61^{4} + \left(17 a^{2} + 12 a + 13\right)\cdot 61^{5} + \left(52 a^{2} + 16 a + 25\right)\cdot 61^{6} + \left(45 a^{2} + 53 a + 8\right)\cdot 61^{7} + \left(25 a^{2} + 33 a + 11\right)\cdot 61^{8} + \left(35 a^{2} + 20 a + 35\right)\cdot 61^{9} + \left(17 a^{2} + 21 a + 28\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 a^{2} + a + 45 + \left(56 a^{2} + 26 a + 4\right)\cdot 61 + \left(2 a^{2} + 33 a + 4\right)\cdot 61^{2} + \left(35 a^{2} + a + 30\right)\cdot 61^{3} + \left(37 a^{2} + 45 a + 47\right)\cdot 61^{4} + \left(45 a^{2} + 59 a + 42\right)\cdot 61^{5} + \left(42 a^{2} + 24 a + 41\right)\cdot 61^{6} + \left(13 a^{2} + 3 a\right)\cdot 61^{7} + \left(35 a^{2} + 48 a + 35\right)\cdot 61^{8} + \left(59 a^{2} + 29 a + 5\right)\cdot 61^{9} + \left(45 a^{2} + 57 a + 39\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 48\cdot 61 + 18\cdot 61^{2} + 60\cdot 61^{3} + 51\cdot 61^{4} + 5\cdot 61^{5} + 50\cdot 61^{6} + 58\cdot 61^{7} + 22\cdot 61^{8} + 13\cdot 61^{9} + 53\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 + 51\cdot 61 + 58\cdot 61^{2} + 32\cdot 61^{3} + 6\cdot 61^{4} + 18\cdot 61^{5} + 13\cdot 61^{6} + 33\cdot 61^{7} + 52\cdot 61^{8} + 34\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 58 + 52\cdot 61 + 49\cdot 61^{2} + 36\cdot 61^{3} + 42\cdot 61^{4} + 21\cdot 61^{5} + 61^{6} + 43\cdot 61^{7} + 34\cdot 61^{8} + 7\cdot 61^{9} + 32\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 15 a^{2} + 37 a + 57 + \left(14 a^{2} + a + 28\right)\cdot 61 + \left(18 a^{2} + 43 a + 11\right)\cdot 61^{2} + \left(20 a^{2} + 48 a + 22\right)\cdot 61^{3} + \left(6 a^{2} + 12 a + 42\right)\cdot 61^{4} + \left(59 a^{2} + 41 a + 3\right)\cdot 61^{5} + \left(18 a^{2} + 39 a + 1\right)\cdot 61^{6} + \left(a^{2} + 44 a + 4\right)\cdot 61^{7} + \left(35 a^{2} + 58 a + 12\right)\cdot 61^{8} + \left(21 a^{2} + 48 a + 20\right)\cdot 61^{9} + \left(35 a^{2} + a + 36\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 57 a^{2} + 3 a + 9 + \left(14 a^{2} + 51 a + 32\right)\cdot 61 + \left(53 a^{2} + 11 a + 32\right)\cdot 61^{2} + \left(8 a^{2} + 34 a + 9\right)\cdot 61^{3} + \left(18 a^{2} + 58 a + 16\right)\cdot 61^{4} + \left(3 a^{2} + 49 a + 48\right)\cdot 61^{5} + \left(18 a^{2} + 36 a + 57\right)\cdot 61^{6} + \left(4 a^{2} + 30 a + 17\right)\cdot 61^{7} + \left(16 a^{2} + 8 a + 25\right)\cdot 61^{8} + \left(31 a^{2} + 8 a + 4\right)\cdot 61^{9} + \left(10 a^{2} + 32 a + 22\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,7)(3,5)(4,6)(8,9)$ |
| $(1,5,4,9,6,3,8,7,2)$ |
| $(1,8,9)(2,3,4)(5,7,6)$ |
| $(2,3,4)(5,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,4)(2,9)(3,8)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,9,8)(2,4,3)(5,6,7)$ | $-3$ |
| $3$ | $3$ | $(1,9,8)(2,3,4)$ | $0$ |
| $3$ | $3$ | $(1,8,9)(2,4,3)$ | $0$ |
| $9$ | $6$ | $(1,3,9,4,8,2)(5,6)$ | $0$ |
| $9$ | $6$ | $(1,2,8,4,9,3)(5,6)$ | $0$ |
| $6$ | $9$ | $(1,5,4,9,6,3,8,7,2)$ | $0$ |
| $6$ | $9$ | $(1,4,6,8,2,5,9,3,7)$ | $0$ |
| $6$ | $9$ | $(1,2,6,8,3,5,9,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
