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:
| $r_{ 1 }$ |
$=$ |
$ 3 a + 43 + \left(58 a + 148\right)\cdot 151 + \left(49 a + 52\right)\cdot 151^{2} + \left(129 a + 106\right)\cdot 151^{3} + \left(108 a + 148\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 a + 76 + \left(8 a + 37\right)\cdot 151 + \left(104 a + 23\right)\cdot 151^{2} + \left(108 a + 76\right)\cdot 151^{3} + \left(150 a + 133\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 148 a + 49 + \left(92 a + 110\right)\cdot 151 + \left(101 a + 93\right)\cdot 151^{2} + \left(21 a + 13\right)\cdot 151^{3} + \left(42 a + 86\right)\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 123 + 50\cdot 151 + 86\cdot 151^{2} + 128\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 103 a + 21 + \left(142 a + 6\right)\cdot 151 + \left(46 a + 72\right)\cdot 151^{2} + \left(42 a + 38\right)\cdot 151^{3} + 24\cdot 151^{4} +O\left(151^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 144 + 99\cdot 151 + 124\cdot 151^{2} + 89\cdot 151^{3} + 35\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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$8$ |
$8$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
$-1$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
$0$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
