Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 11.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{3} + 4 x + 64 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41 a^{2} + 59 a + 19 + \left(62 a^{2} + 2 a + 28\right)\cdot 71 + \left(45 a^{2} + 40 a + 13\right)\cdot 71^{2} + \left(19 a^{2} + 33 a + 19\right)\cdot 71^{3} + \left(54 a^{2} + 68 a + 43\right)\cdot 71^{4} + \left(11 a^{2} + 61 a + 65\right)\cdot 71^{5} + \left(46 a^{2} + 3 a + 32\right)\cdot 71^{6} + \left(20 a^{2} + 53 a + 35\right)\cdot 71^{7} + \left(65 a^{2} + 66 a + 46\right)\cdot 71^{8} + \left(40 a^{2} + 26 a + 8\right)\cdot 71^{9} + \left(57 a^{2} + 54 a + 4\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a^{2} + 59 a + 17 + \left(22 a^{2} + 23 a + 34\right)\cdot 71 + \left(4 a^{2} + 65 a + 28\right)\cdot 71^{2} + \left(69 a^{2} + 18 a + 7\right)\cdot 71^{3} + \left(30 a^{2} + 30 a + 50\right)\cdot 71^{4} + \left(64 a^{2} + 8 a + 25\right)\cdot 71^{5} + \left(28 a^{2} + 33 a + 25\right)\cdot 71^{6} + \left(2 a^{2} + 3 a + 44\right)\cdot 71^{7} + \left(37 a^{2} + 63 a + 58\right)\cdot 71^{8} + \left(50 a^{2} + 6 a + 45\right)\cdot 71^{9} + \left(8 a^{2} + 43 a + 34\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 a^{2} + 47 a + 27 + \left(17 a^{2} + 41 a + 50\right)\cdot 71 + \left(46 a^{2} + 57 a + 61\right)\cdot 71^{2} + \left(54 a^{2} + 5 a + 17\right)\cdot 71^{3} + \left(21 a^{2} + 16 a + 51\right)\cdot 71^{4} + \left(42 a + 58\right)\cdot 71^{5} + \left(4 a^{2} + 20 a + 38\right)\cdot 71^{6} + \left(13 a^{2} + 55 a + 62\right)\cdot 71^{7} + \left(7 a^{2} + 37 a + 9\right)\cdot 71^{8} + \left(47 a^{2} + 20 a + 25\right)\cdot 71^{9} + \left(41 a^{2} + 19 a + 56\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a^{2} + 36 a + 38 + \left(61 a^{2} + 26 a + 2\right)\cdot 71 + \left(49 a^{2} + 44 a + 24\right)\cdot 71^{2} + \left(67 a^{2} + 31 a + 5\right)\cdot 71^{3} + \left(65 a^{2} + 57 a + 27\right)\cdot 71^{4} + \left(58 a^{2} + 37 a + 49\right)\cdot 71^{5} + \left(20 a^{2} + 46 a + 12\right)\cdot 71^{6} + \left(37 a^{2} + 33 a + 56\right)\cdot 71^{7} + \left(69 a^{2} + 37 a + 57\right)\cdot 71^{8} + \left(53 a^{2} + 23 a + 19\right)\cdot 71^{9} + \left(42 a^{2} + 68 a + 59\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a^{2} + 12 a + 33 + \left(69 a^{2} + 18 a + 41\right)\cdot 71 + \left(21 a^{2} + 5 a + 4\right)\cdot 71^{2} + \left(25 a^{2} + 34 a + 9\right)\cdot 71^{3} + \left(64 a^{2} + 34 a + 68\right)\cdot 71^{4} + \left(31 a^{2} + 43 a + 9\right)\cdot 71^{5} + \left(56 a^{2} + 26 a + 4\right)\cdot 71^{6} + \left(6 a^{2} + 41 a + 56\right)\cdot 71^{7} + \left(38 a^{2} + 64 a + 37\right)\cdot 71^{8} + \left(8 a^{2} + 19 a + 28\right)\cdot 71^{9} + \left(5 a^{2} + 54 a + 1\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 + 26\cdot 71 + 20\cdot 71^{2} + 17\cdot 71^{3} + 40\cdot 71^{4} + 11\cdot 71^{5} + 15\cdot 71^{6} + 69\cdot 71^{7} + 32\cdot 71^{8} + 11\cdot 71^{9} + 59\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 a^{2} + 54 + \left(50 a^{2} + 29 a + 61\right)\cdot 71 + \left(44 a^{2} + 17\right)\cdot 71^{2} + \left(47 a^{2} + 18 a + 21\right)\cdot 71^{3} + \left(46 a^{2} + 6 a + 68\right)\cdot 71^{4} + \left(45 a^{2} + 19 a + 22\right)\cdot 71^{5} + \left(56 a^{2} + 11 a + 28\right)\cdot 71^{6} + \left(61 a^{2} + 26 a + 13\right)\cdot 71^{7} + \left(66 a^{2} + 14 a + 67\right)\cdot 71^{8} + \left(11 a^{2} + 44 a + 13\right)\cdot 71^{9} + \left(57 a^{2} + 44 a + 69\right)\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 33 + 41\cdot 71 + 16\cdot 71^{2} + 52\cdot 71^{3} + 35\cdot 71^{4} + 34\cdot 71^{5} + 41\cdot 71^{6} + 51\cdot 71^{7} + 71^{8} + 54\cdot 71^{9} + 35\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 30 + 68\cdot 71 + 25\cdot 71^{2} + 63\cdot 71^{3} + 41\cdot 71^{4} + 5\cdot 71^{5} + 14\cdot 71^{6} + 37\cdot 71^{7} + 42\cdot 71^{8} + 5\cdot 71^{9} + 35\cdot 71^{10} +O\left(71^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3)(2,8)(5,6)(7,9)$ |
| $(1,3,4)(2,7,5)$ |
| $(2,7,5)(6,8,9)$ |
| $(1,2,6,3,5,8,4,7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,9)(2,5)(3,8)(4,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,4)(2,5,7)(6,8,9)$ |
$-3$ |
| $3$ |
$3$ |
$(1,3,4)(2,7,5)$ |
$0$ |
| $3$ |
$3$ |
$(1,4,3)(2,5,7)$ |
$0$ |
| $9$ |
$6$ |
$(1,8,4,9,3,6)(2,5)$ |
$0$ |
| $9$ |
$6$ |
$(1,6,3,9,4,8)(2,5)$ |
$0$ |
| $6$ |
$9$ |
$(1,2,6,3,5,8,4,7,9)$ |
$0$ |
| $6$ |
$9$ |
$(1,6,5,4,9,2,3,8,7)$ |
$0$ |
| $6$ |
$9$ |
$(1,7,9,3,2,6,4,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
