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:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 8 + \left(74 a + 34\right)\cdot 97 + \left(47 a + 22\right)\cdot 97^{2} + \left(83 a + 39\right)\cdot 97^{3} + \left(38 a + 10\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 52 a + 62 + \left(19 a + 3\right)\cdot 97 + \left(35 a + 55\right)\cdot 97^{2} + \left(55 a + 60\right)\cdot 97^{3} + \left(25 a + 29\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 93 + 66\cdot 97 + 79\cdot 97^{2} + 27\cdot 97^{3} + 71\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 + 35\cdot 97 + 67\cdot 97^{2} + 7\cdot 97^{3} + 20\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a + 17 + \left(77 a + 68\right)\cdot 97 + \left(61 a + 70\right)\cdot 97^{2} + \left(41 a + 80\right)\cdot 97^{3} + \left(71 a + 96\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 a + 34 + \left(22 a + 82\right)\cdot 97 + \left(49 a + 92\right)\cdot 97^{2} + \left(13 a + 74\right)\cdot 97^{3} + \left(58 a + 62\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(3,5)$ | $0$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
| $4$ | $3$ | $(1,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $1$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
