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:
| $r_{ 1 }$ |
$=$ |
$ 109 + 68\cdot 229 + 208\cdot 229^{2} + 16\cdot 229^{3} + 137\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 223 a + 16 + \left(77 a + 166\right)\cdot 229 + \left(80 a + 206\right)\cdot 229^{2} + \left(132 a + 39\right)\cdot 229^{3} + \left(160 a + 42\right)\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 185 + 5\cdot 229 + 56\cdot 229^{2} + 76\cdot 229^{3} + 172\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 10 + \left(151 a + 21\right)\cdot 229 + \left(148 a + 209\right)\cdot 229^{2} + \left(96 a + 91\right)\cdot 229^{3} + \left(68 a + 70\right)\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 217 + 22\cdot 229 + 70\cdot 229^{2} + 211\cdot 229^{3} + 102\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 156 + 152\cdot 229 + 114\cdot 229^{2} + 204\cdot 229^{3} + 146\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 224 + 20\cdot 229 + 51\cdot 229^{2} + 46\cdot 229^{3} + 15\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 |
| $1$ | $1$ | $()$ | $15$ |
| $21$ | $2$ | $(1,2)$ | $-5$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $70$ | $3$ | $(1,2,3)$ | $3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $1$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $0$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
