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