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