Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 239 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 239 }$: $ x^{2} + 237 x + 7 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 6 + \left(145 a + 28\right)\cdot 239 + \left(209 a + 121\right)\cdot 239^{2} + \left(204 a + 32\right)\cdot 239^{3} + \left(233 a + 77\right)\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 217 a + 199 + \left(92 a + 213\right)\cdot 239 + \left(144 a + 26\right)\cdot 239^{2} + \left(64 a + 112\right)\cdot 239^{3} + \left(60 a + 149\right)\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 232 a + 20 + \left(93 a + 72\right)\cdot 239 + \left(29 a + 156\right)\cdot 239^{2} + \left(34 a + 232\right)\cdot 239^{3} + \left(5 a + 100\right)\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 134 + 93\cdot 239 + 122\cdot 239^{2} + 117\cdot 239^{3} + 200\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 155 + \left(146 a + 182\right)\cdot 239 + \left(94 a + 222\right)\cdot 239^{2} + \left(174 a + 96\right)\cdot 239^{3} + \left(178 a + 205\right)\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 + 235\cdot 239 + 68\cdot 239^{2} + 41\cdot 239^{3} + 195\cdot 239^{4} +O\left(239^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 179 + 130\cdot 239 + 237\cdot 239^{2} + 83\cdot 239^{3} + 27\cdot 239^{4} +O\left(239^{ 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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$14$ |
| $21$ |
$2$ |
$(1,2)$ |
$-6$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-2$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$2$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$2$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$-1$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$2$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$1$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$0$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$-1$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
