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:
| $r_{ 1 }$ |
$=$ |
$ 25 a + 41 + \left(26 a + 50\right)\cdot 97 + \left(26 a + 33\right)\cdot 97^{2} + \left(58 a + 2\right)\cdot 97^{3} + \left(72 a + 34\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 18 + \left(8 a + 91\right)\cdot 97 + \left(12 a + 73\right)\cdot 97^{2} + \left(82 a + 75\right)\cdot 97^{3} + \left(40 a + 49\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 60 a + 78 + \left(60 a + 13\right)\cdot 97 + \left(63 a + 83\right)\cdot 97^{2} + \left(85 a + 53\right)\cdot 97^{3} + \left(83 a + 27\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 37 a + 41 + \left(36 a + 14\right)\cdot 97 + \left(33 a + 86\right)\cdot 97^{2} + \left(11 a + 75\right)\cdot 97^{3} + \left(13 a + 25\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 72 a + 66 + \left(70 a + 51\right)\cdot 97 + \left(70 a + 33\right)\cdot 97^{2} + \left(38 a + 34\right)\cdot 97^{3} + \left(24 a + 48\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 67 a + 48 + \left(88 a + 69\right)\cdot 97 + \left(84 a + 77\right)\cdot 97^{2} + \left(14 a + 48\right)\cdot 97^{3} + \left(56 a + 8\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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
