Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 157 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 157 }$: $ x^{2} + 152 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 73 + \left(57 a + 146\right)\cdot 157 + \left(57 a + 39\right)\cdot 157^{2} + \left(60 a + 47\right)\cdot 157^{3} + \left(35 a + 61\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 119 a + 73 + \left(87 a + 134\right)\cdot 157 + \left(66 a + 54\right)\cdot 157^{2} + 10\cdot 157^{3} + \left(10 a + 125\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 a + 40 + \left(69 a + 140\right)\cdot 157 + \left(90 a + 142\right)\cdot 157^{2} + \left(156 a + 102\right)\cdot 157^{3} + \left(146 a + 17\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 131 + 134\cdot 157 + 50\cdot 157^{2} + 103\cdot 157^{3} + 78\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 130 a + 51 + \left(99 a + 91\right)\cdot 157 + \left(99 a + 112\right)\cdot 157^{2} + \left(96 a + 134\right)\cdot 157^{3} + \left(121 a + 20\right)\cdot 157^{4} +O\left(157^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 103 + 137\cdot 157 + 69\cdot 157^{2} + 72\cdot 157^{3} + 10\cdot 157^{4} +O\left(157^{ 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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
