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