Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 277 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 277 }$: $ x^{2} + 274 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 8 + \left(3 a + 205\right)\cdot 277 + \left(145 a + 227\right)\cdot 277^{2} + \left(170 a + 240\right)\cdot 277^{3} + \left(139 a + 186\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 232 a + 121 + \left(208 a + 203\right)\cdot 277 + \left(147 a + 104\right)\cdot 277^{2} + \left(240 a + 32\right)\cdot 277^{3} + \left(73 a + 182\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 275 a + 14 + \left(273 a + 212\right)\cdot 277 + \left(131 a + 105\right)\cdot 277^{2} + \left(106 a + 53\right)\cdot 277^{3} + \left(137 a + 158\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 a + 263 + \left(68 a + 43\right)\cdot 277 + \left(129 a + 62\right)\cdot 277^{2} + \left(36 a + 52\right)\cdot 277^{3} + \left(203 a + 163\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 90 a + 78 + \left(86 a + 137\right)\cdot 277 + \left(107 a + 47\right)\cdot 277^{2} + \left(161 a + 176\right)\cdot 277^{3} + \left(244 a + 199\right)\cdot 277^{4} +O\left(277^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 187 a + 71 + \left(190 a + 29\right)\cdot 277 + \left(169 a + 6\right)\cdot 277^{2} + \left(115 a + 276\right)\cdot 277^{3} + \left(32 a + 217\right)\cdot 277^{4} +O\left(277^{ 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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$9$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $15$ |
$2$ |
$(1,2)$ |
$-3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
