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