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:
| $r_{ 1 }$ |
$=$ |
$ 7 + 4\cdot 71 + 26\cdot 71^{2} + 56\cdot 71^{3} + 49\cdot 71^{4} + 21\cdot 71^{5} + 33\cdot 71^{6} + 25\cdot 71^{7} + 53\cdot 71^{8} + 48\cdot 71^{9} + 63\cdot 71^{10} + 54\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 36 a^{2} + 27 a + 67 + \left(13 a^{2} + 27 a + 44\right)\cdot 71 + \left(34 a^{2} + 29 a + 65\right)\cdot 71^{2} + \left(5 a^{2} + 69 a + 22\right)\cdot 71^{3} + \left(28 a^{2} + 5 a + 45\right)\cdot 71^{4} + \left(53 a^{2} + 42 a + 45\right)\cdot 71^{5} + \left(38 a^{2} + 49 a + 18\right)\cdot 71^{6} + \left(69 a^{2} + 36 a + 62\right)\cdot 71^{7} + \left(35 a^{2} + 39 a + 27\right)\cdot 71^{8} + \left(7 a^{2} + 7 a + 11\right)\cdot 71^{9} + \left(68 a^{2} + 70 a + 41\right)\cdot 71^{10} + \left(56 a^{2} + 31 a + 59\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 54 + 6\cdot 71 + 17\cdot 71^{2} + 54\cdot 71^{3} + 38\cdot 71^{4} + 47\cdot 71^{5} + 31\cdot 71^{6} + 48\cdot 71^{7} + 68\cdot 71^{8} + 56\cdot 71^{9} + 2\cdot 71^{10} + 56\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a^{2} + 67 a + 15 + \left(52 a^{2} + 68 a + 32\right)\cdot 71 + \left(58 a^{2} + 47 a + 62\right)\cdot 71^{2} + \left(20 a^{2} + 28 a + 51\right)\cdot 71^{3} + \left(3 a^{2} + 16 a + 22\right)\cdot 71^{4} + \left(33 a^{2} + 70 a + 60\right)\cdot 71^{5} + \left(24 a^{2} + a + 51\right)\cdot 71^{6} + \left(16 a^{2} + 54 a + 23\right)\cdot 71^{7} + \left(13 a^{2} + 70 a + 50\right)\cdot 71^{8} + \left(3 a^{2} + 19 a + 60\right)\cdot 71^{9} + \left(28 a^{2} + 68 a + 23\right)\cdot 71^{10} + \left(68 a^{2} + 53 a + 6\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 + 32\cdot 71^{2} + 17\cdot 71^{3} + 29\cdot 71^{4} + 20\cdot 71^{5} + 16\cdot 71^{6} + 37\cdot 71^{8} + 47\cdot 71^{9} + 10\cdot 71^{10} + 55\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 56 a^{2} + 25 a + 2 + \left(29 a^{2} + 41\right)\cdot 71 + \left(53 a^{2} + 20 a + 69\right)\cdot 71^{2} + \left(58 a^{2} + 28 a + 22\right)\cdot 71^{3} + \left(30 a^{2} + 7 a + 5\right)\cdot 71^{4} + \left(64 a^{2} + 68 a + 4\right)\cdot 71^{5} + \left(13 a^{2} + 24 a + 47\right)\cdot 71^{6} + \left(2 a^{2} + 14 a + 24\right)\cdot 71^{7} + \left(68 a^{2} + 39 a + 42\right)\cdot 71^{8} + \left(28 a^{2} + 27 a + 68\right)\cdot 71^{9} + \left(63 a^{2} + 28 a + 4\right)\cdot 71^{10} + \left(25 a^{2} + 6 a + 24\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 2 a^{2} + 8 a + 9 + \left(14 a^{2} + 40 a + 1\right)\cdot 71 + \left(9 a^{2} + 34 a + 25\right)\cdot 71^{2} + \left(23 a^{2} + 57 a + 34\right)\cdot 71^{3} + \left(6 a^{2} + 58 a + 54\right)\cdot 71^{4} + \left(45 a^{2} + 4 a + 68\right)\cdot 71^{5} + \left(43 a^{2} + 50 a + 31\right)\cdot 71^{6} + \left(26 a^{2} + 61 a + 27\right)\cdot 71^{7} + \left(60 a^{2} + 46 a + 10\right)\cdot 71^{8} + \left(19 a^{2} + 42 a + 34\right)\cdot 71^{9} + \left(51 a^{2} + 20 a + 38\right)\cdot 71^{10} + \left(24 a^{2} + 50 a + 55\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 50 a^{2} + 19 a + 57 + \left(27 a^{2} + 43 a + 11\right)\cdot 71 + \left(54 a^{2} + 21 a + 1\right)\cdot 71^{2} + \left(6 a^{2} + 44 a + 50\right)\cdot 71^{3} + \left(12 a^{2} + 57 a + 2\right)\cdot 71^{4} + \left(24 a^{2} + 31 a + 15\right)\cdot 71^{5} + \left(18 a^{2} + 67 a + 35\right)\cdot 71^{6} + \left(70 a^{2} + 19 a + 40\right)\cdot 71^{7} + \left(37 a^{2} + 63 a + 9\right)\cdot 71^{8} + \left(34 a^{2} + 35 a + 36\right)\cdot 71^{9} + \left(10 a^{2} + 43 a + 29\right)\cdot 71^{10} + \left(59 a^{2} + 32 a + 65\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 47 a^{2} + 67 a + 58 + \left(4 a^{2} + 32 a + 70\right)\cdot 71 + \left(3 a^{2} + 59 a + 55\right)\cdot 71^{2} + \left(27 a^{2} + 55 a + 44\right)\cdot 71^{3} + \left(61 a^{2} + 66 a + 35\right)\cdot 71^{4} + \left(63 a^{2} + 66 a\right)\cdot 71^{5} + \left(2 a^{2} + 18 a + 18\right)\cdot 71^{6} + \left(28 a^{2} + 26 a + 31\right)\cdot 71^{7} + \left(68 a^{2} + 24 a + 55\right)\cdot 71^{8} + \left(47 a^{2} + 8 a + 61\right)\cdot 71^{9} + \left(62 a^{2} + 53 a + 68\right)\cdot 71^{10} + \left(48 a^{2} + 37 a + 48\right)\cdot 71^{11} +O\left(71^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,8,7)(2,4,5)(3,6,9)$ |
| $(2,6,8)(4,7,9)$ |
| $(2,4)(6,9)(7,8)$ |
| $(1,5,3)(2,6,8)(4,9,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(2,4)(6,9)(7,8)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,6,8)(4,9,7)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,8,6)(4,7,9)$ | $-3 \zeta_{3} - 3$ |
| $6$ | $3$ | $(1,8,7)(2,4,5)(3,6,9)$ | $0$ |
| $6$ | $3$ | $(1,2,7)(3,8,9)(4,5,6)$ | $0$ |
| $6$ | $3$ | $(1,6,7)(2,9,3)(4,5,8)$ | $0$ |
| $6$ | $3$ | $(2,6,8)(4,7,9)$ | $0$ |
| $9$ | $6$ | $(1,5,3)(2,9,8,4,6,7)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,3,5)(2,7,6,4,8,9)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
