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:
| $r_{ 1 }$ |
$=$ |
$ 17 a^{2} + 24 a + 10 + \left(8 a^{2} + 9 a + 36\right)\cdot 61 + \left(60 a^{2} + 47 a + 13\right)\cdot 61^{2} + \left(48 a^{2} + 33 a + 44\right)\cdot 61^{3} + \left(13 a^{2} + 37 a + 13\right)\cdot 61^{4} + \left(53 a^{2} + 36 a + 1\right)\cdot 61^{5} + \left(18 a^{2} + 36 a + 56\right)\cdot 61^{6} + \left(35 a^{2} + a + 39\right)\cdot 61^{7} + \left(8 a^{2} + 14 a + 6\right)\cdot 61^{8} + \left(40 a^{2} + 51 a + 25\right)\cdot 61^{9} + \left(29 a^{2} + 37 a + 59\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 a^{2} + 54 a + 14 + \left(42 a^{2} + 40 a + 60\right)\cdot 61 + \left(39 a^{2} + 11 a + 27\right)\cdot 61^{2} + \left(40 a^{2} + 46 a + 26\right)\cdot 61^{3} + \left(23 a^{2} + 46 a + 27\right)\cdot 61^{4} + \left(24 a^{2} + 20 a + 32\right)\cdot 61^{5} + \left(14 a^{2} + 56 a + 16\right)\cdot 61^{6} + \left(42 a^{2} + 6 a + 38\right)\cdot 61^{7} + \left(42 a^{2} + 8 a + 23\right)\cdot 61^{8} + \left(6 a^{2} + 10 a + 7\right)\cdot 61^{9} + \left(47 a^{2} + 20 a + 11\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 + 10\cdot 61 + 44\cdot 61^{2} + 35\cdot 61^{3} + 2\cdot 61^{4} + 19\cdot 61^{5} + 51\cdot 61^{6} + 21\cdot 61^{7} + 16\cdot 61^{8} + 32\cdot 61^{9} + 55\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 56\cdot 61 + 35\cdot 61^{2} + 59\cdot 61^{3} + 33\cdot 61^{4} + 40\cdot 61^{5} + 26\cdot 61^{6} + 31\cdot 61^{7} + 41\cdot 61^{8} + 34\cdot 61^{9} + 39\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 a^{2} + 16 a + 53 + \left(37 a^{2} + 52 a + 51\right)\cdot 61 + \left(46 a^{2} + 14 a + 31\right)\cdot 61^{2} + \left(7 a^{2} + 36 a + 14\right)\cdot 61^{3} + \left(24 a^{2} + 10 a + 21\right)\cdot 61^{4} + \left(52 a + 39\right)\cdot 61^{5} + \left(22 a^{2} + 58 a + 9\right)\cdot 61^{6} + \left(27 a^{2} + 57 a + 23\right)\cdot 61^{7} + \left(14 a^{2} + 17 a + 54\right)\cdot 61^{8} + \left(47 a^{2} + 32 a + 37\right)\cdot 61^{9} + \left(58 a^{2} + 38 a + 32\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 41 a^{2} + 14 a + 13 + \left(5 a^{2} + 45 a + 53\right)\cdot 61 + \left(48 a^{2} + 11 a + 46\right)\cdot 61^{2} + \left(30 a^{2} + 17 a\right)\cdot 61^{3} + \left(17 a^{2} + 4 a + 19\right)\cdot 61^{4} + \left(60 a^{2} + 52 a + 37\right)\cdot 61^{5} + \left(11 a^{2} + 26 a + 25\right)\cdot 61^{6} + \left(26 a^{2} + 11 a + 24\right)\cdot 61^{7} + \left(4 a^{2} + 53 a + 48\right)\cdot 61^{8} + \left(17 a^{2} + 5 a + 55\right)\cdot 61^{9} + \left(17 a^{2} + 10 a + 54\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 a^{2} + 54 a + 48 + \left(13 a^{2} + 35 a + 27\right)\cdot 61 + \left(34 a^{2} + 37 a + 2\right)\cdot 61^{2} + \left(50 a^{2} + 58 a + 32\right)\cdot 61^{3} + \left(19 a^{2} + 9 a + 9\right)\cdot 61^{4} + \left(37 a^{2} + 49 a + 52\right)\cdot 61^{5} + \left(34 a^{2} + 38 a + 49\right)\cdot 61^{6} + \left(53 a^{2} + 42 a + 9\right)\cdot 61^{7} + \left(13 a^{2} + 60 a + 11\right)\cdot 61^{8} + \left(37 a^{2} + 44 a + 48\right)\cdot 61^{9} + \left(57 a^{2} + 30 a + 39\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 14 + 43\cdot 61 + 33\cdot 61^{2} + 32\cdot 61^{3} + 59\cdot 61^{4} + 10\cdot 61^{5} + 48\cdot 61^{6} + 4\cdot 61^{7} + 20\cdot 61^{8} + 3\cdot 61^{9} + 36\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 57 a^{2} + 21 a + 34 + \left(14 a^{2} + 60 a + 26\right)\cdot 61 + \left(15 a^{2} + 59 a + 7\right)\cdot 61^{2} + \left(4 a^{2} + 51 a + 59\right)\cdot 61^{3} + \left(23 a^{2} + 12 a + 56\right)\cdot 61^{4} + \left(7 a^{2} + 33 a + 10\right)\cdot 61^{5} + \left(20 a^{2} + 26 a + 21\right)\cdot 61^{6} + \left(59 a^{2} + a + 50\right)\cdot 61^{7} + \left(37 a^{2} + 29 a + 21\right)\cdot 61^{8} + \left(34 a^{2} + 38 a + 60\right)\cdot 61^{9} + \left(33 a^{2} + 45 a + 36\right)\cdot 61^{10} +O\left(61^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,8)(3,7)(4,6)(5,9)$ |
| $(2,7,6)(3,4,8)$ |
| $(1,4,7)(2,9,8)(3,6,5)$ |
| $(1,5,9)(2,6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,2)(3,4)(5,6)(7,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,9,5)(2,6,7)(3,4,8)$ |
$-3$ |
| $3$ |
$3$ |
$(1,5,9)(2,6,7)$ |
$0$ |
| $3$ |
$3$ |
$(1,9,5)(2,7,6)$ |
$0$ |
| $6$ |
$3$ |
$(1,4,7)(2,9,8)(3,6,5)$ |
$0$ |
| $6$ |
$3$ |
$(1,7,4)(2,8,9)(3,5,6)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,7)(2,9,4)(5,8,6)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,9,2,5,7)(3,4)$ |
$0$ |
| $9$ |
$6$ |
$(1,7,5,2,9,6)(3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
