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:
| $r_{ 1 }$ |
$=$ |
$ 68 a^{2} + 18 a + 19 + \left(52 a^{2} + 57 a + 52\right)\cdot 71 + \left(69 a^{2} + 41 a + 54\right)\cdot 71^{2} + \left(31 a^{2} + 58 a + 54\right)\cdot 71^{3} + \left(4 a^{2} + 68 a + 15\right)\cdot 71^{4} + \left(67 a^{2} + 49 a + 3\right)\cdot 71^{5} + \left(23 a^{2} + 52 a + 33\right)\cdot 71^{6} + \left(9 a^{2} + 29 a + 67\right)\cdot 71^{7} + \left(52 a^{2} + 23 a + 6\right)\cdot 71^{8} + \left(68 a^{2} + 19 a + 9\right)\cdot 71^{9} + \left(41 a^{2} + 10 a + 49\right)\cdot 71^{10} + \left(14 a^{2} + 52 a + 23\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 63 a^{2} + 30 a + 53 + \left(42 a^{2} + 60 a + 1\right)\cdot 71 + \left(42 a^{2} + 66 a + 6\right)\cdot 71^{2} + \left(64 a^{2} + 68 a + 47\right)\cdot 71^{3} + \left(24 a^{2} + 48 a + 46\right)\cdot 71^{4} + \left(31 a^{2} + 9 a + 2\right)\cdot 71^{5} + \left(16 a^{2} + 30 a + 13\right)\cdot 71^{6} + \left(6 a^{2} + 38 a + 59\right)\cdot 71^{7} + \left(42 a^{2} + 69 a + 3\right)\cdot 71^{8} + \left(57 a^{2} + 69 a + 3\right)\cdot 71^{9} + \left(58 a^{2} + 33 a + 23\right)\cdot 71^{10} + \left(5 a^{2} + 16 a\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 a^{2} + 12 a + 70 + \left(26 a^{2} + 5 a + 47\right)\cdot 71 + \left(22 a^{2} + 19 a + 37\right)\cdot 71^{2} + \left(35 a^{2} + 70 a + 21\right)\cdot 71^{3} + \left(56 a^{2} + 33 a + 37\right)\cdot 71^{4} + \left(23 a^{2} + 56 a + 21\right)\cdot 71^{5} + \left(19 a^{2} + 12 a + 12\right)\cdot 71^{6} + \left(18 a^{2} + 26 a + 12\right)\cdot 71^{7} + \left(56 a + 32\right)\cdot 71^{8} + \left(67 a^{2} + 29 a + 26\right)\cdot 71^{9} + \left(57 a^{2} + 52 a\right)\cdot 71^{10} + \left(25 a^{2} + 3 a + 57\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a^{2} + 23 a + 9 + \left(46 a^{2} + 24 a + 34\right)\cdot 71 + \left(29 a^{2} + 33 a + 42\right)\cdot 71^{2} + \left(45 a^{2} + 14 a + 19\right)\cdot 71^{3} + \left(41 a^{2} + 24 a + 20\right)\cdot 71^{4} + \left(43 a^{2} + 11 a + 35\right)\cdot 71^{5} + \left(30 a^{2} + 59 a + 3\right)\cdot 71^{6} + \left(55 a^{2} + 2 a + 1\right)\cdot 71^{7} + \left(47 a^{2} + 49 a + 19\right)\cdot 71^{8} + \left(15 a^{2} + 52 a + 33\right)\cdot 71^{9} + \left(41 a^{2} + 26 a + 23\right)\cdot 71^{10} + \left(50 a^{2} + 2 a + 1\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 a^{2} + 24 a + 23 + \left(21 a^{2} + 17 a + 11\right)\cdot 71 + \left(60 a^{2} + 41 a + 44\right)\cdot 71^{2} + \left(37 a^{2} + 68 a + 4\right)\cdot 71^{3} + \left(37 a^{2} + 26 a + 34\right)\cdot 71^{4} + \left(41 a^{2} + 4 a + 21\right)\cdot 71^{5} + \left(35 a^{2} + 35 a + 8\right)\cdot 71^{6} + \left(40 a^{2} + 18 a + 24\right)\cdot 71^{7} + \left(27 a^{2} + 7 a + 10\right)\cdot 71^{8} + \left(2 a^{2} + 27 a + 67\right)\cdot 71^{9} + \left(54 a^{2} + 7 a + 60\right)\cdot 71^{10} + \left(41 a^{2} + 43 a + 4\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a^{2} + 35 a + 25 + \left(22 a^{2} + 48 a + 13\right)\cdot 71 + \left(59 a^{2} + 10 a + 65\right)\cdot 71^{2} + \left(68 a^{2} + 3 a + 39\right)\cdot 71^{3} + \left(47 a^{2} + 10 a + 14\right)\cdot 71^{4} + \left(5 a^{2} + 10 a + 44\right)\cdot 71^{5} + \left(16 a^{2} + 23 a + 3\right)\cdot 71^{6} + \left(12 a^{2} + 26 a + 67\right)\cdot 71^{7} + \left(43 a^{2} + 7 a + 51\right)\cdot 71^{8} + \left(a^{2} + 14 a + 17\right)\cdot 71^{9} + \left(30 a^{2} + 11 a + 44\right)\cdot 71^{10} + \left(3 a^{2} + 24 a + 44\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 + 4\cdot 71 + 12\cdot 71^{2} + 5\cdot 71^{3} + 21\cdot 71^{4} + 17\cdot 71^{5} + 41\cdot 71^{6} + 66\cdot 71^{7} + 43\cdot 71^{8} + 29\cdot 71^{9} + 51\cdot 71^{10} + 25\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 55 + 67\cdot 71 + 33\cdot 71^{2} + 40\cdot 71^{3} + 41\cdot 71^{4} + 26\cdot 71^{5} + 50\cdot 71^{6} + 71^{7} + 49\cdot 71^{8} + 44\cdot 71^{9} + 57\cdot 71^{10} + 62\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 65 + 50\cdot 71 + 58\cdot 71^{2} + 50\cdot 71^{3} + 52\cdot 71^{4} + 40\cdot 71^{5} + 47\cdot 71^{6} + 55\cdot 71^{7} + 66\cdot 71^{8} + 52\cdot 71^{9} + 44\cdot 71^{10} + 63\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,8,2,6,7,4,3,9)$ |
| $(1,2,4)(3,5,6)(7,9,8)$ |
| $(2,4)(3,9)(5,7)(6,8)$ |
| $(3,6,5)(7,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,8)(2,9)(4,7)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,4)(3,5,6)(7,9,8)$ | $-3$ |
| $3$ | $3$ | $(1,2,4)(7,8,9)$ | $0$ |
| $3$ | $3$ | $(1,4,2)(7,9,8)$ | $0$ |
| $9$ | $6$ | $(1,7,2,8,4,9)(5,6)$ | $0$ |
| $9$ | $6$ | $(1,9,4,8,2,7)(5,6)$ | $0$ |
| $6$ | $9$ | $(1,5,8,2,6,7,4,3,9)$ | $0$ |
| $6$ | $9$ | $(1,8,6,4,9,5,2,7,3)$ | $0$ |
| $6$ | $9$ | $(1,3,8,2,5,7,4,6,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
