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:
| $r_{ 1 }$ |
$=$ |
$ 28 + 89\cdot 97 + 96\cdot 97^{2} + 63\cdot 97^{3} + 62\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 57 + \left(68 a + 7\right)\cdot 97 + \left(45 a + 96\right)\cdot 97^{2} + \left(77 a + 52\right)\cdot 97^{3} + \left(86 a + 65\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 82 + 17\cdot 97 + 75\cdot 97^{2} + 84\cdot 97^{3} + 48\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 a + 2 + \left(28 a + 34\right)\cdot 97 + \left(51 a + 73\right)\cdot 97^{2} + \left(19 a + 84\right)\cdot 97^{3} + \left(10 a + 74\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 95 + \left(68 a + 2\right)\cdot 97 + \left(4 a + 55\right)\cdot 97^{2} + \left(6 a + 1\right)\cdot 97^{3} + \left(48 a + 47\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 a + 27 + \left(28 a + 42\right)\cdot 97 + \left(92 a + 88\right)\cdot 97^{2} + \left(90 a + 2\right)\cdot 97^{3} + \left(48 a + 89\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ |
| $6$ | $2$ | $(2,4)$ | $0$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,4)$ | $-2$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,4,3)$ | $1$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
