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:
| $r_{ 1 }$ |
$=$ |
$ 72 a + 14 + \left(78 a + 137\right)\cdot 173 + \left(107 a + 96\right)\cdot 173^{2} + \left(60 a + 60\right)\cdot 173^{3} + \left(152 a + 77\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 a + 164 + \left(57 a + 59\right)\cdot 173 + \left(39 a + 170\right)\cdot 173^{2} + \left(149 a + 85\right)\cdot 173^{3} + \left(170 a + 126\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 118 a + 38 + \left(115 a + 61\right)\cdot 173 + \left(133 a + 97\right)\cdot 173^{2} + \left(23 a + 124\right)\cdot 173^{3} + \left(2 a + 141\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 13\cdot 173 + 56\cdot 173^{2} + 156\cdot 173^{3} + 59\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 162 + 81\cdot 173 + 157\cdot 173^{2} + 141\cdot 173^{3} + 77\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 140 + 132\cdot 173 + 11\cdot 173^{2} + 100\cdot 173^{3} + 101\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 101 a + 129 + \left(94 a + 32\right)\cdot 173 + \left(65 a + 102\right)\cdot 173^{2} + \left(112 a + 22\right)\cdot 173^{3} + \left(20 a + 107\right)\cdot 173^{4} +O\left(173^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
