Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{3} + 4 x + 64 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 a^{2} + 70 a + 59 + \left(22 a^{2} + 6 a + 10\right)\cdot 71 + \left(49 a^{2} + 54 a + 15\right)\cdot 71^{2} + \left(18 a^{2} + 27 a + 65\right)\cdot 71^{3} + \left(69 a^{2} + 37 a + 45\right)\cdot 71^{4} + \left(23 a^{2} + a + 31\right)\cdot 71^{5} + \left(53 a^{2} + 61 a + 39\right)\cdot 71^{6} + \left(59 a^{2} + 47 a + 52\right)\cdot 71^{7} + \left(10 a^{2} + 68 a + 21\right)\cdot 71^{8} + \left(63 a^{2} + 50 a + 51\right)\cdot 71^{9} + \left(56 a^{2} + 52 a\right)\cdot 71^{10} + \left(47 a^{2} + 52 a + 50\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a^{2} + 36 a + 44 + \left(64 a^{2} + 2\right)\cdot 71 + \left(34 a^{2} + 51 a + 24\right)\cdot 71^{2} + \left(44 a^{2} + 27 a + 39\right)\cdot 71^{3} + \left(70 a^{2} + 59 a + 49\right)\cdot 71^{4} + \left(20 a^{2} + 63 a + 23\right)\cdot 71^{5} + \left(17 a^{2} + 63 a + 14\right)\cdot 71^{6} + \left(25 a^{2} + 30 a + 55\right)\cdot 71^{7} + \left(20 a^{2} + 60 a + 70\right)\cdot 71^{8} + \left(49 a^{2} + 56 a + 37\right)\cdot 71^{9} + \left(21 a^{2} + 66 a + 1\right)\cdot 71^{10} + \left(50 a^{2} + 4 a + 33\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a^{2} + 36 a + 5 + \left(55 a^{2} + 63 a + 50\right)\cdot 71 + \left(57 a^{2} + 36 a + 37\right)\cdot 71^{2} + \left(7 a^{2} + 15 a + 12\right)\cdot 71^{3} + \left(2 a^{2} + 45 a + 56\right)\cdot 71^{4} + \left(26 a^{2} + 5 a + 60\right)\cdot 71^{5} + \left(17 a + 63\right)\cdot 71^{6} + \left(57 a^{2} + 63 a + 68\right)\cdot 71^{7} + \left(39 a^{2} + 12 a + 27\right)\cdot 71^{8} + \left(29 a^{2} + 34 a + 56\right)\cdot 71^{9} + \left(63 a^{2} + 22 a + 41\right)\cdot 71^{10} + \left(43 a^{2} + 13 a + 39\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a^{2} + 44 a + 48 + \left(9 a^{2} + 34 a + 8\right)\cdot 71 + \left(10 a^{2} + 34 a + 43\right)\cdot 71^{2} + \left(6 a^{2} + 46 a + 58\right)\cdot 71^{3} + \left(28 a^{2} + 54 a + 59\right)\cdot 71^{4} + \left(53 a^{2} + 63 a + 52\right)\cdot 71^{5} + \left(62 a^{2} + 48 a + 19\right)\cdot 71^{6} + \left(64 a^{2} + 33 a + 21\right)\cdot 71^{7} + \left(5 a^{2} + 2 a + 17\right)\cdot 71^{8} + \left(48 a^{2} + 60 a + 4\right)\cdot 71^{9} + \left(19 a^{2} + 22 a + 47\right)\cdot 71^{10} + \left(25 a^{2} + 7 a + 27\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a^{2} + 17 a + 66 + \left(18 a^{2} + 46 a + 55\right)\cdot 71 + \left(44 a^{2} + 65 a + 15\right)\cdot 71^{2} + \left(57 a^{2} + 63 a + 30\right)\cdot 71^{3} + \left(15 a^{2} + 21 a + 3\right)\cdot 71^{4} + \left(41 a^{2} + 33 a + 44\right)\cdot 71^{5} + \left(65 a^{2} + 4 a + 3\right)\cdot 71^{6} + \left(63 a^{2} + 23 a + 66\right)\cdot 71^{7} + \left(19 a^{2} + 4 a + 30\right)\cdot 71^{8} + \left(58 a^{2} + 20 a + 31\right)\cdot 71^{9} + \left(26 a^{2} + 13 a + 42\right)\cdot 71^{10} + \left(4 a^{2} + 20 a + 66\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a^{2} + 10 a + 2 + \left(43 a^{2} + 61 a + 27\right)\cdot 71 + \left(16 a^{2} + 41 a + 60\right)\cdot 71^{2} + \left(7 a^{2} + 31 a + 37\right)\cdot 71^{3} + \left(27 a^{2} + 65 a + 33\right)\cdot 71^{4} + \left(47 a^{2} + 44 a + 60\right)\cdot 71^{5} + \left(13 a^{2} + 17 a + 30\right)\cdot 71^{6} + \left(13 a^{2} + 14 a + 25\right)\cdot 71^{7} + \left(45 a^{2} + 64 a + 27\right)\cdot 71^{8} + \left(35 a^{2} + 61 a + 18\right)\cdot 71^{9} + \left(24 a^{2} + 34 a + 36\right)\cdot 71^{10} + \left(41 a^{2} + 43 a + 70\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 44\cdot 71 + 54\cdot 71^{2} + 58\cdot 71^{3} + 39\cdot 71^{4} + 23\cdot 71^{5} + 60\cdot 71^{6} + 66\cdot 71^{7} + 32\cdot 71^{8} + 69\cdot 71^{9} + 60\cdot 71^{10} + 49\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 33 + 46\cdot 71 + 32\cdot 71^{2} + 16\cdot 71^{3} + 40\cdot 71^{4} + 24\cdot 71^{5} + 47\cdot 71^{6} + 44\cdot 71^{7} + 9\cdot 71^{8} + 70\cdot 71^{9} + 28\cdot 71^{10} + 4\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 66 + 37\cdot 71 + 36\cdot 71^{3} + 26\cdot 71^{4} + 33\cdot 71^{5} + 4\cdot 71^{6} + 25\cdot 71^{7} + 45\cdot 71^{8} + 15\cdot 71^{9} + 24\cdot 71^{10} + 13\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,7,5)(2,9,6)(3,8,4)$ |
| $(1,2)(4,5)(7,8)$ |
| $(1,2,3)(4,5,6)(7,9,8)$ |
| $(4,8)(5,7)(6,9)$ |
| $(4,6,5)(7,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,2)(5,6)(8,9)$ |
$0$ |
| $9$ |
$2$ |
$(4,8)(5,7)(6,9)$ |
$-2$ |
| $9$ |
$2$ |
$(1,7)(2,8)(3,9)(4,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)(4,5,6)(7,9,8)$ |
$-3$ |
| $6$ |
$3$ |
$(1,5,9)(2,6,8)(3,4,7)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,2)(7,9,8)$ |
$0$ |
| $12$ |
$3$ |
$(1,7,5)(2,9,6)(3,8,4)$ |
$0$ |
| $18$ |
$6$ |
$(1,8,5,2,9,6)(3,7,4)$ |
$0$ |
| $18$ |
$6$ |
$(1,2,3)(4,7,6,8,5,9)$ |
$1$ |
| $18$ |
$6$ |
$(1,8,3,7,2,9)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
