Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{2} + 149 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 35 + 63\cdot 151 + 55\cdot 151^{2} + 17\cdot 151^{3} + 5\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 + 55\cdot 151 + 108\cdot 151^{2} + 97\cdot 151^{3} + 125\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 145 a + 111 + \left(30 a + 49\right)\cdot 151 + \left(124 a + 68\right)\cdot 151^{2} + \left(112 a + 51\right)\cdot 151^{3} + \left(19 a + 130\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 122 a + 32 + \left(39 a + 29\right)\cdot 151 + \left(36 a + 102\right)\cdot 151^{2} + \left(122 a + 113\right)\cdot 151^{3} + \left(14 a + 113\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 a + 99 + \left(120 a + 117\right)\cdot 151 + \left(26 a + 134\right)\cdot 151^{2} + \left(38 a + 1\right)\cdot 151^{3} + \left(131 a + 57\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 125 + \left(111 a + 137\right)\cdot 151 + \left(114 a + 134\right)\cdot 151^{2} + \left(28 a + 19\right)\cdot 151^{3} + \left(136 a + 21\right)\cdot 151^{4} +O\left(151^{ 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.
