Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 263 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 263 }$: $ x^{2} + 261 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 133 a + 38 + \left(48 a + 27\right)\cdot 263 + \left(110 a + 166\right)\cdot 263^{2} + \left(72 a + 96\right)\cdot 263^{3} + \left(257 a + 184\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 a + 25 + \left(52 a + 181\right)\cdot 263 + \left(191 a + 46\right)\cdot 263^{2} + \left(195 a + 91\right)\cdot 263^{3} + \left(74 a + 47\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 210 + 153\cdot 263 + 79\cdot 263^{2} + 86\cdot 263^{3} + 235\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 130 a + 41 + \left(214 a + 254\right)\cdot 263 + \left(152 a + 74\right)\cdot 263^{2} + \left(190 a + 131\right)\cdot 263^{3} + \left(5 a + 100\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 a + 189 + \left(162 a + 62\right)\cdot 263 + \left(123 a + 111\right)\cdot 263^{2} + \left(63 a + 44\right)\cdot 263^{3} + \left(137 a + 136\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 232 a + 87 + \left(210 a + 254\right)\cdot 263 + \left(71 a + 113\right)\cdot 263^{2} + \left(67 a + 28\right)\cdot 263^{3} + \left(188 a + 1\right)\cdot 263^{4} +O\left(263^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 258 a + 199 + \left(100 a + 118\right)\cdot 263 + \left(139 a + 196\right)\cdot 263^{2} + \left(199 a + 47\right)\cdot 263^{3} + \left(125 a + 84\right)\cdot 263^{4} +O\left(263^{ 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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
