Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 173 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 173 }$: $ x^{2} + 169 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 112 + 143\cdot 173 + 144\cdot 173^{2} + 71\cdot 173^{3} + 99\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 148 a + 133 + \left(28 a + 79\right)\cdot 173 + \left(64 a + 106\right)\cdot 173^{2} + \left(20 a + 4\right)\cdot 173^{3} + \left(10 a + 55\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 33 + \left(144 a + 47\right)\cdot 173 + \left(108 a + 161\right)\cdot 173^{2} + \left(152 a + 21\right)\cdot 173^{3} + \left(162 a + 75\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 98 a + 128 + \left(56 a + 112\right)\cdot 173 + \left(156 a + 162\right)\cdot 173^{2} + \left(28 a + 156\right)\cdot 173^{3} + \left(85 a + 129\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 113 + 67\cdot 173 + 77\cdot 173^{2} + 147\cdot 173^{3} + 63\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 75 a + 1 + \left(116 a + 68\right)\cdot 173 + \left(16 a + 39\right)\cdot 173^{2} + \left(144 a + 116\right)\cdot 173^{3} + \left(87 a + 95\right)\cdot 173^{4} +O\left(173^{ 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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
