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:
| $r_{ 1 }$ |
$=$ |
$ 72 a + 22 + \left(3 a + 17\right)\cdot 97 + \left(95 a + 76\right)\cdot 97^{2} + \left(42 a + 82\right)\cdot 97^{3} + \left(7 a + 39\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 a + 19 + \left(55 a + 30\right)\cdot 97 + \left(20 a + 75\right)\cdot 97^{2} + \left(69 a + 14\right)\cdot 97^{3} + \left(a + 57\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 91 + 92\cdot 97 + 66\cdot 97^{2} + 84\cdot 97^{3} + 49\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 94 + \left(93 a + 45\right)\cdot 97 + \left(a + 70\right)\cdot 97^{2} + \left(54 a + 30\right)\cdot 97^{3} + \left(89 a + 4\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 + 86\cdot 97 + 58\cdot 97^{2} + 14\cdot 97^{3} + 53\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 87 + \left(41 a + 17\right)\cdot 97 + \left(76 a + 40\right)\cdot 97^{2} + \left(27 a + 63\right)\cdot 97^{3} + \left(95 a + 86\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
