Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 88 a + 69 + \left(96 a + 34\right)\cdot 97 + \left(71 a + 70\right)\cdot 97^{2} + \left(73 a + 14\right)\cdot 97^{3} + \left(64 a + 51\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 60 + 43\cdot 97 + \left(25 a + 45\right)\cdot 97^{2} + \left(23 a + 16\right)\cdot 97^{3} + \left(32 a + 42\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 62 a + 10 + \left(77 a + 3\right)\cdot 97 + \left(31 a + 33\right)\cdot 97^{2} + \left(93 a + 42\right)\cdot 97^{3} + \left(41 a + 81\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 96\cdot 97 + 37\cdot 97^{2} + 86\cdot 97^{3} + 19\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 72 + \left(19 a + 18\right)\cdot 97 + \left(65 a + 84\right)\cdot 97^{2} + \left(3 a + 6\right)\cdot 97^{3} + \left(55 a + 30\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 94\cdot 97 + 19\cdot 97^{2} + 27\cdot 97^{3} + 66\cdot 97^{4} +O\left(97^{ 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.
