Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 397 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 397 }$: $ x^{2} + 392 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 282 + 305\cdot 397 + 79\cdot 397^{2} + 80\cdot 397^{3} + 121\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 301 + 164\cdot 397 + 135\cdot 397^{2} + 362\cdot 397^{3} + 56\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 131 a + 373 + \left(208 a + 189\right)\cdot 397 + \left(321 a + 333\right)\cdot 397^{2} + 3 a\cdot 397^{3} + \left(49 a + 79\right)\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 364 a + 282 + \left(96 a + 51\right)\cdot 397 + \left(71 a + 119\right)\cdot 397^{2} + \left(311 a + 179\right)\cdot 397^{3} + \left(127 a + 341\right)\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 266 a + 234 + \left(188 a + 306\right)\cdot 397 + \left(75 a + 144\right)\cdot 397^{2} + \left(393 a + 95\right)\cdot 397^{3} + \left(347 a + 320\right)\cdot 397^{4} +O\left(397^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 117 + \left(300 a + 172\right)\cdot 397 + \left(325 a + 378\right)\cdot 397^{2} + \left(85 a + 75\right)\cdot 397^{3} + \left(269 a + 272\right)\cdot 397^{4} +O\left(397^{ 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 |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $15$ |
$2$ |
$(1,2)$ |
$-1$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
