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:
| $r_{ 1 }$ |
$=$ |
$ 88 + 65\cdot 151 + 70\cdot 151^{2} + 123\cdot 151^{3} + 34\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 57 a + 107 + \left(99 a + 53\right)\cdot 151 + \left(93 a + 9\right)\cdot 151^{2} + \left(55 a + 123\right)\cdot 151^{3} + \left(150 a + 85\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 48 + \left(8 a + 21\right)\cdot 151 + \left(91 a + 117\right)\cdot 151^{2} + \left(126 a + 50\right)\cdot 151^{3} + \left(43 a + 92\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 94 a + 70 + \left(51 a + 44\right)\cdot 151 + \left(57 a + 97\right)\cdot 151^{2} + \left(95 a + 140\right)\cdot 151^{3} + 28\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 92\cdot 151 + 18\cdot 151^{2} + 104\cdot 151^{3} + 6\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 138 a + 74 + \left(142 a + 24\right)\cdot 151 + \left(59 a + 140\right)\cdot 151^{2} + \left(24 a + 61\right)\cdot 151^{3} + \left(107 a + 53\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
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.
