Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 229 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 229 }$: $ x^{2} + 228 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 216 + 92\cdot 229 + 185\cdot 229^{2} + 35\cdot 229^{3} + 154\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 138 a + 90 + \left(112 a + 111\right)\cdot 229 + \left(92 a + 46\right)\cdot 229^{2} + \left(108 a + 137\right)\cdot 229^{3} + \left(61 a + 182\right)\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 a + 116 + \left(184 a + 227\right)\cdot 229 + \left(226 a + 174\right)\cdot 229^{2} + \left(125 a + 32\right)\cdot 229^{3} + \left(3 a + 16\right)\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 91 a + 228 + \left(116 a + 85\right)\cdot 229 + \left(136 a + 26\right)\cdot 229^{2} + \left(120 a + 153\right)\cdot 229^{3} + \left(167 a + 135\right)\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 53 + \left(200 a + 145\right)\cdot 229 + \left(199 a + 132\right)\cdot 229^{2} + \left(161 a + 102\right)\cdot 229^{3} + \left(86 a + 75\right)\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 204 a + 141 + \left(44 a + 157\right)\cdot 229 + \left(2 a + 217\right)\cdot 229^{2} + \left(103 a + 160\right)\cdot 229^{3} + \left(225 a + 122\right)\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 208 a + 74 + \left(28 a + 95\right)\cdot 229 + \left(29 a + 132\right)\cdot 229^{2} + \left(67 a + 64\right)\cdot 229^{3} + 142 a\cdot 229^{4} +O\left(229^{ 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 |
| $1$ | $1$ | $()$ | $6$ |
| $21$ | $2$ | $(1,2)$ | $4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $-1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
