Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 281 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 281 }$: $ x^{2} + 280 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 61 a + 175 + \left(258 a + 178\right)\cdot 281 + \left(100 a + 227\right)\cdot 281^{2} + \left(33 a + 54\right)\cdot 281^{3} + \left(255 a + 137\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 220 a + 236 + \left(22 a + 94\right)\cdot 281 + \left(180 a + 70\right)\cdot 281^{2} + \left(247 a + 268\right)\cdot 281^{3} + \left(25 a + 77\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 254 + 271\cdot 281 + 42\cdot 281^{2} + 150\cdot 281^{3} + 263\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 272 a + 225 + \left(30 a + 77\right)\cdot 281 + \left(62 a + 53\right)\cdot 281^{2} + \left(245 a + 252\right)\cdot 281^{3} + \left(154 a + 223\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 216 + \left(250 a + 117\right)\cdot 281 + \left(218 a + 84\right)\cdot 281^{2} + \left(35 a + 154\right)\cdot 281^{3} + \left(126 a + 133\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 156 a + 212 + \left(240 a + 8\right)\cdot 281 + \left(241 a + 41\right)\cdot 281^{2} + \left(179 a + 153\right)\cdot 281^{3} + \left(96 a + 185\right)\cdot 281^{4} +O\left(281^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 125 a + 87 + \left(40 a + 93\right)\cdot 281 + \left(39 a + 42\right)\cdot 281^{2} + \left(101 a + 91\right)\cdot 281^{3} + \left(184 a + 102\right)\cdot 281^{4} +O\left(281^{ 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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
