Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 151 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 151 }$: $ x^{2} + 149 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 35 a + 17 + \left(114 a + 46\right)\cdot 151 + \left(90 a + 39\right)\cdot 151^{2} + \left(95 a + 93\right)\cdot 151^{3} + \left(16 a + 119\right)\cdot 151^{4} + \left(91 a + 62\right)\cdot 151^{5} +O\left(151^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 101 + \left(116 a + 42\right)\cdot 151 + \left(132 a + 12\right)\cdot 151^{2} + \left(43 a + 142\right)\cdot 151^{3} + \left(57 a + 140\right)\cdot 151^{4} + \left(124 a + 131\right)\cdot 151^{5} +O\left(151^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 116 a + 87 + \left(36 a + 88\right)\cdot 151 + \left(60 a + 106\right)\cdot 151^{2} + \left(55 a + 42\right)\cdot 151^{3} + \left(134 a + 57\right)\cdot 151^{4} + \left(59 a + 77\right)\cdot 151^{5} +O\left(151^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 + 73\cdot 151 + 98\cdot 151^{2} + 93\cdot 151^{3} + 115\cdot 151^{4} + 118\cdot 151^{5} +O\left(151^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 145 a + 113 + \left(34 a + 117\right)\cdot 151 + \left(18 a + 10\right)\cdot 151^{2} + \left(107 a + 97\right)\cdot 151^{3} + \left(93 a + 60\right)\cdot 151^{4} + \left(26 a + 21\right)\cdot 151^{5} +O\left(151^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 70 + 84\cdot 151 + 34\cdot 151^{2} + 135\cdot 151^{3} + 109\cdot 151^{4} + 40\cdot 151^{5} +O\left(151^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $8$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
