Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 17.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{3} + 4 x + 64 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 a^{2} + 52 a + 43 + \left(a^{2} + 39 a + 43\right)\cdot 71 + \left(56 a^{2} + a + 13\right)\cdot 71^{2} + \left(a^{2} + 68 a + 24\right)\cdot 71^{3} + \left(38 a^{2} + 59 a + 41\right)\cdot 71^{4} + \left(33 a^{2} + 54 a\right)\cdot 71^{5} + \left(24 a^{2} + 50 a + 68\right)\cdot 71^{6} + \left(66 a^{2} + 43 a + 53\right)\cdot 71^{7} + \left(23 a^{2} + 12 a + 13\right)\cdot 71^{8} + \left(4 a^{2} + 17 a + 2\right)\cdot 71^{9} + \left(34 a^{2} + 51 a + 23\right)\cdot 71^{10} + \left(51 a^{2} + 25 a + 5\right)\cdot 71^{11} + \left(23 a^{2} + 52 a + 12\right)\cdot 71^{12} + \left(25 a^{2} + 9 a + 28\right)\cdot 71^{13} + \left(8 a^{2} + 15 a + 18\right)\cdot 71^{14} + \left(6 a^{2} + 54 a + 63\right)\cdot 71^{15} + \left(25 a^{2} + 38 a + 41\right)\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 70 a^{2} + 55 a + 29 + \left(20 a^{2} + 36 a + 1\right)\cdot 71 + \left(54 a^{2} + 34 a + 9\right)\cdot 71^{2} + \left(20 a^{2} + 49 a + 51\right)\cdot 71^{3} + \left(62 a^{2} + 25 a + 58\right)\cdot 71^{4} + \left(31 a^{2} + 32 a + 19\right)\cdot 71^{5} + \left(10 a^{2} + 48 a + 54\right)\cdot 71^{6} + \left(31 a^{2} + 15 a + 54\right)\cdot 71^{7} + \left(25 a^{2} + 30 a + 17\right)\cdot 71^{8} + \left(32 a^{2} + 43 a + 53\right)\cdot 71^{9} + \left(4 a^{2} + 68 a + 38\right)\cdot 71^{10} + \left(35 a^{2} + 42 a + 32\right)\cdot 71^{11} + \left(62 a^{2} + 40 a + 44\right)\cdot 71^{12} + \left(29 a^{2} + 28 a + 16\right)\cdot 71^{13} + \left(26 a^{2} + 16 a + 19\right)\cdot 71^{14} + \left(39 a^{2} + 31 a + 57\right)\cdot 71^{15} + \left(37 a^{2} + 40 a + 27\right)\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a^{2} + 53 a + 5 + \left(27 a^{2} + 48 a + 47\right)\cdot 71 + \left(31 a^{2} + 41 a + 55\right)\cdot 71^{2} + \left(14 a^{2} + 45 a + 60\right)\cdot 71^{3} + \left(12 a^{2} + 40 a + 66\right)\cdot 71^{4} + \left(32 a^{2} + 26 a + 2\right)\cdot 71^{5} + \left(15 a^{2} + 47 a + 33\right)\cdot 71^{6} + \left(44 a^{2} + 36 a + 41\right)\cdot 71^{7} + \left(26 a^{2} + 32 a + 41\right)\cdot 71^{8} + \left(37 a^{2} + 66 a + 62\right)\cdot 71^{9} + \left(26 a^{2} + 42 a + 7\right)\cdot 71^{10} + \left(50 a^{2} + 66 a + 3\right)\cdot 71^{11} + \left(41 a^{2} + 16\right)\cdot 71^{12} + \left(17 a^{2} + 17 a + 35\right)\cdot 71^{13} + \left(52 a^{2} + 12 a + 45\right)\cdot 71^{14} + \left(11 a^{2} + a + 46\right)\cdot 71^{15} + \left(10 a^{2} + 37 a + 64\right)\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 a^{2} + 35 a + 23 + \left(48 a^{2} + 65 a + 4\right)\cdot 71 + \left(31 a^{2} + 34 a + 67\right)\cdot 71^{2} + \left(48 a^{2} + 24 a + 53\right)\cdot 71^{3} + \left(41 a^{2} + 56 a + 3\right)\cdot 71^{4} + \left(5 a^{2} + 54 a + 68\right)\cdot 71^{5} + \left(36 a^{2} + 42 a + 27\right)\cdot 71^{6} + \left(44 a^{2} + 11 a + 19\right)\cdot 71^{7} + \left(21 a^{2} + 28 a + 31\right)\cdot 71^{8} + \left(34 a^{2} + 10 a + 58\right)\cdot 71^{9} + \left(32 a^{2} + 22 a + 18\right)\cdot 71^{10} + \left(55 a^{2} + 2 a + 63\right)\cdot 71^{11} + \left(55 a^{2} + 49 a + 2\right)\cdot 71^{12} + \left(15 a^{2} + 32 a + 50\right)\cdot 71^{13} + \left(36 a^{2} + 39 a + 21\right)\cdot 71^{14} + \left(25 a^{2} + 56 a + 20\right)\cdot 71^{15} + \left(8 a^{2} + 62 a + 68\right)\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 4\cdot 71 + 30\cdot 71^{2} + 30\cdot 71^{3} + 20\cdot 71^{4} + 48\cdot 71^{5} + 44\cdot 71^{6} + 52\cdot 71^{7} + 67\cdot 71^{8} + 69\cdot 71^{9} + 55\cdot 71^{10} + 62\cdot 71^{11} + 71^{12} + 22\cdot 71^{13} + 39\cdot 71^{14} + 46\cdot 71^{15} + 7\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 54 a^{2} + 9 a + 19 + \left(42 a^{2} + 64 a + 64\right)\cdot 71 + \left(33 a^{2} + 15 a + 37\right)\cdot 71^{2} + \left(21 a^{2} + 47 a + 8\right)\cdot 71^{3} + \left(29 a^{2} + 40 a + 65\right)\cdot 71^{4} + \left(58 a^{2} + 70 a + 1\right)\cdot 71^{5} + \left(18 a^{2} + 5 a + 42\right)\cdot 71^{6} + \left(37 a^{2} + 31 a + 46\right)\cdot 71^{7} + \left(57 a^{2} + 22 a + 5\right)\cdot 71^{8} + \left(a^{2} + 48 a + 15\right)\cdot 71^{9} + \left(20 a^{2} + 60 a + 14\right)\cdot 71^{10} + \left(3 a^{2} + 65 a + 43\right)\cdot 71^{11} + \left(30 a^{2} + 48 a + 8\right)\cdot 71^{12} + \left(36 a^{2} + 41 a + 38\right)\cdot 71^{13} + \left(19 a^{2} + 31 a + 5\right)\cdot 71^{14} + \left(6 a^{2} + 45 a + 32\right)\cdot 71^{15} + \left(52 a^{2} + 9 a + 34\right)\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 6 + 69\cdot 71 + 29\cdot 71^{3} + 7\cdot 71^{4} + 7\cdot 71^{5} + 14\cdot 71^{6} + 45\cdot 71^{7} + 20\cdot 71^{8} + 20\cdot 71^{9} + 64\cdot 71^{10} + 46\cdot 71^{11} + 55\cdot 71^{12} + 71^{13} + 66\cdot 71^{14} + 30\cdot 71^{15} + 4\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 40 + 26\cdot 71 + 34\cdot 71^{2} + 28\cdot 71^{3} + 49\cdot 71^{4} + 33\cdot 71^{5} + 28\cdot 71^{6} + 3\cdot 71^{7} + 8\cdot 71^{8} + 49\cdot 71^{9} + 58\cdot 71^{10} + 40\cdot 71^{11} + 26\cdot 71^{12} + 58\cdot 71^{13} + 45\cdot 71^{14} + 19\cdot 71^{15} + 21\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 57 a^{2} + 9 a + 27 + \left(29 a + 23\right)\cdot 71 + \left(6 a^{2} + 13 a + 35\right)\cdot 71^{2} + \left(35 a^{2} + 49 a + 68\right)\cdot 71^{3} + \left(29 a^{2} + 60 a + 41\right)\cdot 71^{4} + \left(51 a^{2} + 44 a + 30\right)\cdot 71^{5} + \left(36 a^{2} + 17 a + 42\right)\cdot 71^{6} + \left(60 a^{2} + 3 a + 37\right)\cdot 71^{7} + \left(57 a^{2} + 16 a + 6\right)\cdot 71^{8} + \left(31 a^{2} + 27 a + 24\right)\cdot 71^{9} + \left(24 a^{2} + 38 a + 2\right)\cdot 71^{10} + \left(17 a^{2} + 9 a + 57\right)\cdot 71^{11} + \left(70 a^{2} + 21 a + 44\right)\cdot 71^{12} + \left(16 a^{2} + 12 a + 33\right)\cdot 71^{13} + \left(70 a^{2} + 27 a + 22\right)\cdot 71^{14} + \left(52 a^{2} + 24 a + 38\right)\cdot 71^{15} + \left(8 a^{2} + 24 a + 13\right)\cdot 71^{16} +O\left(71^{ 17 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,4,9,2,6)(5,7,8)$ |
| $(1,2,4)$ |
| $(1,2,4)(3,7,6,8,9,5)$ |
| $(5,7,8)$ |
| $(3,6,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $9$ |
$2$ |
$(1,9)(2,3)(4,6)$ |
$1$ |
$1$ |
| $1$ |
$3$ |
$(1,4,2)(3,9,6)(5,8,7)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $1$ |
$3$ |
$(1,2,4)(3,6,9)(5,7,8)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $3$ |
$3$ |
$(1,2,4)$ |
$-\zeta_{3} - 2$ |
$\zeta_{3} - 1$ |
| $3$ |
$3$ |
$(1,4,2)$ |
$\zeta_{3} - 1$ |
$-\zeta_{3} - 2$ |
| $3$ |
$3$ |
$(1,2,4)(3,6,9)(5,8,7)$ |
$2 \zeta_{3} + 1$ |
$-2 \zeta_{3} - 1$ |
| $3$ |
$3$ |
$(1,4,2)(3,9,6)(5,7,8)$ |
$-2 \zeta_{3} - 1$ |
$2 \zeta_{3} + 1$ |
| $3$ |
$3$ |
$(1,2,4)(3,6,9)$ |
$\zeta_{3} + 2$ |
$-\zeta_{3} + 1$ |
| $3$ |
$3$ |
$(1,4,2)(3,9,6)$ |
$-\zeta_{3} + 1$ |
$\zeta_{3} + 2$ |
| $6$ |
$3$ |
$(1,4,2)(3,6,9)$ |
$0$ |
$0$ |
| $18$ |
$3$ |
$(1,7,9)(2,8,3)(4,5,6)$ |
$0$ |
$0$ |
| $9$ |
$6$ |
$(1,3,4,9,2,6)(5,7,8)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $9$ |
$6$ |
$(1,6,2,9,4,3)(5,8,7)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $9$ |
$6$ |
$(1,9,2,3,4,6)(5,7,8)$ |
$1$ |
$1$ |
| $9$ |
$6$ |
$(1,6,4,3,2,9)(5,8,7)$ |
$1$ |
$1$ |
| $9$ |
$6$ |
$(1,3)(2,6)(4,9)(5,7,8)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $9$ |
$6$ |
$(1,3)(2,6)(4,9)(5,8,7)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $9$ |
$6$ |
$(3,7,6,8,9,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $9$ |
$6$ |
$(3,5,9,8,6,7)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $18$ |
$9$ |
$(1,7,9,4,5,6,2,8,3)$ |
$0$ |
$0$ |
| $18$ |
$9$ |
$(1,9,5,2,3,7,4,6,8)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
