Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{3} + 2 x + 68 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 2\cdot 73 + 45\cdot 73^{2} + 36\cdot 73^{3} + 73^{4} + 12\cdot 73^{5} + 15\cdot 73^{6} + 55\cdot 73^{7} + 35\cdot 73^{8} + 28\cdot 73^{9} +O\left(73^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 65 a^{2} + 20 a + 38 + \left(30 a^{2} + 41 a + 65\right)\cdot 73 + \left(3 a^{2} + 54 a + 28\right)\cdot 73^{2} + \left(a^{2} + 14 a + 1\right)\cdot 73^{3} + \left(65 a^{2} + 60 a + 38\right)\cdot 73^{4} + \left(33 a^{2} + 45 a + 69\right)\cdot 73^{5} + \left(33 a^{2} + 57 a + 68\right)\cdot 73^{6} + \left(55 a^{2} + 69 a\right)\cdot 73^{7} + \left(19 a^{2} + 50 a + 2\right)\cdot 73^{8} + \left(9 a^{2} + 13 a + 61\right)\cdot 73^{9} +O\left(73^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 a^{2} + 46 a + 11 + \left(22 a^{2} + 31 a + 6\right)\cdot 73 + \left(59 a^{2} + 58 a + 6\right)\cdot 73^{2} + \left(10 a^{2} + 19 a + 63\right)\cdot 73^{3} + \left(67 a^{2} + 62 a + 40\right)\cdot 73^{4} + \left(47 a^{2} + 11 a + 39\right)\cdot 73^{5} + \left(26 a^{2} + 39 a + 35\right)\cdot 73^{6} + \left(71 a^{2} + 35 a + 46\right)\cdot 73^{7} + \left(43 a^{2} + 11 a + 58\right)\cdot 73^{8} + \left(11 a^{2} + 39 a + 39\right)\cdot 73^{9} +O\left(73^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 40 + 20\cdot 73 + 35\cdot 73^{2} + 69\cdot 73^{3} + 35\cdot 73^{4} + 29\cdot 73^{5} + 61\cdot 73^{6} + 48\cdot 73^{7} + 67\cdot 73^{8} + 43\cdot 73^{9} +O\left(73^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a^{2} + 14 a + 12 + \left(7 a^{2} + 54 a + 58\right)\cdot 73 + \left(11 a^{2} + 53 a + 14\right)\cdot 73^{2} + \left(9 a^{2} + 13 a + 12\right)\cdot 73^{3} + \left(61 a^{2} + 49 a + 57\right)\cdot 73^{4} + \left(34 a^{2} + 23 a + 70\right)\cdot 73^{5} + \left(68 a^{2} + 28 a + 66\right)\cdot 73^{6} + \left(28 a^{2} + 72 a + 62\right)\cdot 73^{7} + \left(16 a + 24\right)\cdot 73^{8} + \left(19 a^{2} + 25\right)\cdot 73^{9} +O\left(73^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 a^{2} + 3 a + 15 + \left(28 a^{2} + 27 a + 14\right)\cdot 73 + \left(36 a^{2} + 31 a + 24\right)\cdot 73^{2} + \left(48 a^{2} + 36 a + 40\right)\cdot 73^{3} + \left(59 a^{2} + 55 a + 6\right)\cdot 73^{4} + \left(60 a^{2} + 49 a + 8\right)\cdot 73^{5} + \left(45 a^{2} + 47 a + 61\right)\cdot 73^{6} + \left(38 a^{2} + 33 a + 2\right)\cdot 73^{7} + \left(38 a^{2} + 10 a + 27\right)\cdot 73^{8} + \left(13 a^{2} + 23 a + 42\right)\cdot 73^{9} +O\left(73^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 + 50\cdot 73 + 65\cdot 73^{2} + 39\cdot 73^{3} + 35\cdot 73^{4} + 31\cdot 73^{5} + 69\cdot 73^{6} + 41\cdot 73^{7} + 42\cdot 73^{8} +O\left(73^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 72 a^{2} + 39 a + 23 + \left(34 a^{2} + 50 a + 22\right)\cdot 73 + \left(58 a^{2} + 37 a + 29\right)\cdot 73^{2} + \left(62 a^{2} + 44 a + 59\right)\cdot 73^{3} + \left(19 a^{2} + 36 a + 50\right)\cdot 73^{4} + \left(4 a^{2} + 3 a + 5\right)\cdot 73^{5} + \left(44 a^{2} + 60 a + 10\right)\cdot 73^{6} + \left(61 a^{2} + 3 a + 9\right)\cdot 73^{7} + \left(52 a^{2} + 5 a + 46\right)\cdot 73^{8} + \left(44 a^{2} + 59 a + 59\right)\cdot 73^{9} +O\left(73^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 17 a^{2} + 24 a + 47 + \left(21 a^{2} + 14 a + 52\right)\cdot 73 + \left(50 a^{2} + 56 a + 42\right)\cdot 73^{2} + \left(13 a^{2} + 16 a + 42\right)\cdot 73^{3} + \left(19 a^{2} + 28 a + 25\right)\cdot 73^{4} + \left(37 a^{2} + 11 a + 25\right)\cdot 73^{5} + \left(59 a + 49\right)\cdot 73^{6} + \left(36 a^{2} + 3 a + 23\right)\cdot 73^{7} + \left(63 a^{2} + 51 a + 60\right)\cdot 73^{8} + \left(47 a^{2} + 10 a + 63\right)\cdot 73^{9} +O\left(73^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(2,6)(3,8)(5,9)$ |
| $(1,4,7)(3,9,6)$ |
| $(1,4)(2,8)(3,6)$ |
| $(2,8,5)(3,6,9)$ |
| $(1,8,3)(2,6,4)(5,9,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,4)(2,8)(3,6)$ | $0$ |
| $9$ | $2$ | $(2,6)(3,8)(5,9)$ | $-2$ |
| $9$ | $2$ | $(1,4)(2,3)(5,9)(6,8)$ | $0$ |
| $2$ | $3$ | $(1,7,4)(2,8,5)(3,9,6)$ | $-3$ |
| $6$ | $3$ | $(1,8,3)(2,6,4)(5,9,7)$ | $0$ |
| $6$ | $3$ | $(1,4,7)(3,9,6)$ | $0$ |
| $12$ | $3$ | $(1,5,3)(2,9,7)(4,8,6)$ | $0$ |
| $18$ | $6$ | $(1,2,3,4,8,6)(5,9,7)$ | $0$ |
| $18$ | $6$ | $(1,4,7)(2,3,8,9,5,6)$ | $1$ |
| $18$ | $6$ | $(1,7)(2,9,5,6,8,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
