Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$: $ x^{2} + 177 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 128 + 137\cdot 181 + 8\cdot 181^{2} + 150\cdot 181^{3} + 5\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 90 a + 137 + \left(174 a + 180\right)\cdot 181 + \left(89 a + 174\right)\cdot 181^{2} + \left(103 a + 61\right)\cdot 181^{3} + \left(169 a + 173\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 123 a + 141 + \left(17 a + 49\right)\cdot 181 + \left(160 a + 131\right)\cdot 181^{2} + \left(80 a + 14\right)\cdot 181^{3} + \left(38 a + 22\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 91 a + 135 + \left(6 a + 64\right)\cdot 181 + \left(91 a + 179\right)\cdot 181^{2} + \left(77 a + 23\right)\cdot 181^{3} + \left(11 a + 24\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 95 + 112\cdot 181 + 18\cdot 181^{2} + 114\cdot 181^{3} + 41\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a + 90 + \left(163 a + 178\right)\cdot 181 + \left(20 a + 29\right)\cdot 181^{2} + \left(100 a + 178\right)\cdot 181^{3} + \left(142 a + 94\right)\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $-3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
