Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{3} + 2 x + 68 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 65\cdot 73 + 11\cdot 73^{2} + 6\cdot 73^{3} + 73^{4} + 24\cdot 73^{5} + 6\cdot 73^{6} + 27\cdot 73^{7} + 46\cdot 73^{8} + 63\cdot 73^{9} + 72\cdot 73^{10} + 46\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 59\cdot 73 + 16\cdot 73^{2} + 36\cdot 73^{3} + 39\cdot 73^{4} + 60\cdot 73^{5} + 40\cdot 73^{6} + 56\cdot 73^{7} + 48\cdot 73^{8} + 64\cdot 73^{9} + 62\cdot 73^{10} + 49\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a^{2} + 3 a + 28 + \left(4 a^{2} + 70 a + 38\right)\cdot 73 + \left(23 a^{2} + 28 a + 66\right)\cdot 73^{2} + \left(68 a^{2} + 10 a + 24\right)\cdot 73^{3} + \left(32 a^{2} + 48 a + 28\right)\cdot 73^{4} + \left(3 a^{2} + 53 a + 55\right)\cdot 73^{5} + \left(15 a^{2} + 44 a + 72\right)\cdot 73^{6} + \left(18 a^{2} + 36 a + 24\right)\cdot 73^{7} + \left(35 a^{2} + 30 a + 2\right)\cdot 73^{8} + \left(48 a^{2} + 21 a + 11\right)\cdot 73^{9} + \left(8 a^{2} + 37 a + 1\right)\cdot 73^{10} + \left(15 a^{2} + 30 a + 70\right)\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 4 a^{2} + 5 a + 22 + \left(49 a^{2} + 11 a + 53\right)\cdot 73 + \left(53 a^{2} + 69 a + 40\right)\cdot 73^{2} + \left(49 a^{2} + 12 a + 3\right)\cdot 73^{3} + \left(17 a^{2} + 53 a + 14\right)\cdot 73^{4} + \left(31 a^{2} + 42 a + 54\right)\cdot 73^{5} + \left(44 a^{2} + a + 69\right)\cdot 73^{6} + \left(17 a^{2} + 26 a + 68\right)\cdot 73^{7} + \left(61 a^{2} + 59 a + 50\right)\cdot 73^{8} + \left(14 a^{2} + 32 a + 22\right)\cdot 73^{9} + \left(66 a^{2} + 38 a + 42\right)\cdot 73^{10} + \left(45 a^{2} + 57 a + 53\right)\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a^{2} + 18 a + 52 + \left(39 a^{2} + 53 a + 36\right)\cdot 73 + \left(69 a^{2} + 53 a + 55\right)\cdot 73^{2} + \left(48 a^{2} + 19 a + 47\right)\cdot 73^{3} + \left(21 a^{2} + 68 a + 37\right)\cdot 73^{4} + \left(13 a^{2} + 20 a + 68\right)\cdot 73^{5} + \left(35 a^{2} + 26 a + 50\right)\cdot 73^{6} + \left(17 a^{2} + 57 a + 72\right)\cdot 73^{7} + \left(16 a^{2} + 70 a + 49\right)\cdot 73^{8} + \left(23 a^{2} + 51 a + 1\right)\cdot 73^{9} + \left(55 a^{2} + 3 a + 39\right)\cdot 73^{10} + \left(6 a^{2} + 41 a + 34\right)\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 72 a^{2} + 13 a + 64 + \left(56 a^{2} + 18 a + 63\right)\cdot 73 + \left(69 a^{2} + 9 a + 37\right)\cdot 73^{2} + \left(21 a^{2} + 34 a + 39\right)\cdot 73^{3} + \left(35 a^{2} + 34 a + 37\right)\cdot 73^{4} + \left(16 a^{2} + 41 a + 34\right)\cdot 73^{5} + \left(55 a^{2} + 20 a + 35\right)\cdot 73^{6} + \left(20 a^{2} + 52 a + 24\right)\cdot 73^{7} + \left(17 a^{2} + 44 a + 65\right)\cdot 73^{8} + \left(47 a^{2} + 70 a + 65\right)\cdot 73^{9} + \left(34 a^{2} + 61 a + 48\right)\cdot 73^{10} + \left(19 a^{2} + 68 a + 42\right)\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 70 a^{2} + 55 a + 37 + \left(39 a^{2} + 43 a + 65\right)\cdot 73 + \left(22 a^{2} + 67 a + 47\right)\cdot 73^{2} + \left(a^{2} + 25 a + 60\right)\cdot 73^{3} + \left(20 a^{2} + 58 a + 65\right)\cdot 73^{4} + \left(25 a^{2} + 61 a + 21\right)\cdot 73^{5} + \left(46 a^{2} + 50 a + 72\right)\cdot 73^{6} + \left(34 a^{2} + 67 a + 42\right)\cdot 73^{7} + \left(67 a^{2} + 41 a + 10\right)\cdot 73^{8} + \left(10 a^{2} + 42 a + 66\right)\cdot 73^{9} + \left(45 a^{2} + 45 a + 62\right)\cdot 73^{10} + \left(7 a^{2} + 19 a + 26\right)\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 33 a^{2} + 52 a + 33 + \left(29 a^{2} + 22 a + 23\right)\cdot 73 + \left(53 a^{2} + 63 a + 58\right)\cdot 73^{2} + \left(28 a^{2} + 42 a + 20\right)\cdot 73^{3} + \left(18 a^{2} + 29 a + 33\right)\cdot 73^{4} + \left(56 a^{2} + 71 a + 28\right)\cdot 73^{5} + \left(22 a^{2} + a + 34\right)\cdot 73^{6} + \left(37 a^{2} + 52 a + 50\right)\cdot 73^{7} + \left(21 a^{2} + 44 a + 32\right)\cdot 73^{8} + \left(a^{2} + 72 a + 45\right)\cdot 73^{9} + \left(9 a^{2} + 31 a + 1\right)\cdot 73^{10} + \left(51 a^{2} + a + 45\right)\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 65 + 31\cdot 73 + 29\cdot 73^{2} + 52\cdot 73^{3} + 34\cdot 73^{4} + 17\cdot 73^{5} + 55\cdot 73^{6} + 69\cdot 73^{7} + 57\cdot 73^{8} + 23\cdot 73^{9} + 33\cdot 73^{10} + 68\cdot 73^{11} +O\left(73^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,5,6)(2,3,4)(7,9,8)$ |
| $(1,2,9)(4,6,7)$ |
| $(1,6)(2,4)(7,9)$ |
| $(1,2,9)(3,8,5)(4,7,6)$ |
| $(1,2)(3,5)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$0$ |
| $9$ |
$2$ |
$(1,6)(2,4)(7,9)$ |
$-2$ |
| $9$ |
$2$ |
$(1,4)(2,6)(3,5)(7,9)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,9)(3,8,5)(4,7,6)$ |
$-3$ |
| $6$ |
$3$ |
$(1,5,6)(2,3,4)(7,9,8)$ |
$0$ |
| $6$ |
$3$ |
$(3,8,5)(4,6,7)$ |
$0$ |
| $12$ |
$3$ |
$(1,5,7)(2,3,6)(4,9,8)$ |
$0$ |
| $18$ |
$6$ |
$(1,3,6,2,5,4)(7,9,8)$ |
$0$ |
| $18$ |
$6$ |
$(1,4,9,6,2,7)(3,8,5)$ |
$1$ |
| $18$ |
$6$ |
$(1,9)(3,7,8,4,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
