Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 72 a + 38 + \left(30 a + 62\right)\cdot 73 + 58 a\cdot 73^{2} + \left(4 a + 22\right)\cdot 73^{3} + \left(34 a + 24\right)\cdot 73^{4} + \left(57 a + 40\right)\cdot 73^{5} + \left(70 a + 68\right)\cdot 73^{6} + \left(23 a + 35\right)\cdot 73^{7} + \left(72 a + 49\right)\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a + 27 + \left(70 a + 67\right)\cdot 73 + \left(9 a + 56\right)\cdot 73^{2} + \left(29 a + 70\right)\cdot 73^{3} + \left(53 a + 43\right)\cdot 73^{4} + \left(48 a + 26\right)\cdot 73^{5} + \left(16 a + 72\right)\cdot 73^{6} + \left(10 a + 65\right)\cdot 73^{7} + \left(18 a + 50\right)\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 58\cdot 73 + 66\cdot 73^{2} + 58\cdot 73^{3} + 47\cdot 73^{4} + 46\cdot 73^{5} + 41\cdot 73^{6} + 65\cdot 73^{7} + 64\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 35 + \left(42 a + 10\right)\cdot 73 + \left(14 a + 72\right)\cdot 73^{2} + \left(68 a + 50\right)\cdot 73^{3} + \left(38 a + 48\right)\cdot 73^{4} + \left(15 a + 32\right)\cdot 73^{5} + \left(2 a + 4\right)\cdot 73^{6} + \left(49 a + 37\right)\cdot 73^{7} + 23\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 a + 46 + \left(2 a + 5\right)\cdot 73 + \left(63 a + 16\right)\cdot 73^{2} + \left(43 a + 2\right)\cdot 73^{3} + \left(19 a + 29\right)\cdot 73^{4} + \left(24 a + 46\right)\cdot 73^{5} + 56 a\cdot 73^{6} + \left(62 a + 7\right)\cdot 73^{7} + \left(54 a + 22\right)\cdot 73^{8} +O\left(73^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 57 + 14\cdot 73 + 6\cdot 73^{2} + 14\cdot 73^{3} + 25\cdot 73^{4} + 26\cdot 73^{5} + 31\cdot 73^{6} + 7\cdot 73^{7} + 8\cdot 73^{8} +O\left(73^{ 9 }\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 value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $6$ | $2$ | $(2,3)(5,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $1$ |
| $6$ | $4$ | $(1,5,4,2)(3,6)$ | $-1$ |
| $8$ | $6$ | $(1,5,6,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
