Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 80 a + 9 + \left(77 a + 1\right)\cdot 97 + \left(55 a + 11\right)\cdot 97^{2} + \left(8 a + 72\right)\cdot 97^{3} + \left(53 a + 74\right)\cdot 97^{4} + \left(89 a + 78\right)\cdot 97^{5} + \left(29 a + 29\right)\cdot 97^{6} + \left(68 a + 29\right)\cdot 97^{7} +O\left(97^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 + 70\cdot 97 + 90\cdot 97^{2} + 33\cdot 97^{3} + 93\cdot 97^{4} + 70\cdot 97^{5} + 27\cdot 97^{6} + 59\cdot 97^{7} +O\left(97^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 89 + \left(19 a + 95\right)\cdot 97 + \left(41 a + 85\right)\cdot 97^{2} + \left(88 a + 24\right)\cdot 97^{3} + \left(43 a + 22\right)\cdot 97^{4} + \left(7 a + 18\right)\cdot 97^{5} + \left(67 a + 67\right)\cdot 97^{6} + \left(28 a + 67\right)\cdot 97^{7} +O\left(97^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 58 a + 20 + \left(83 a + 84\right)\cdot 97 + \left(57 a + 12\right)\cdot 97^{2} + \left(34 a + 60\right)\cdot 97^{3} + \left(76 a + 27\right)\cdot 97^{4} + \left(42 a + 65\right)\cdot 97^{5} + \left(36 a + 51\right)\cdot 97^{6} + \left(27 a + 4\right)\cdot 97^{7} +O\left(97^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 55 + 26\cdot 97 + 6\cdot 97^{2} + 63\cdot 97^{3} + 3\cdot 97^{4} + 26\cdot 97^{5} + 69\cdot 97^{6} + 37\cdot 97^{7} +O\left(97^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 78 + \left(13 a + 12\right)\cdot 97 + \left(39 a + 84\right)\cdot 97^{2} + \left(62 a + 36\right)\cdot 97^{3} + \left(20 a + 69\right)\cdot 97^{4} + \left(54 a + 31\right)\cdot 97^{5} + \left(60 a + 45\right)\cdot 97^{6} + \left(69 a + 92\right)\cdot 97^{7} +O\left(97^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)$ |
| $(1,3)$ |
| $(4,6)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(2,5)$ | $-1$ |
| $4$ | $3$ | $(1,4,2)(3,6,5)$ | $0$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $0$ |
| $4$ | $6$ | $(1,6,5,3,4,2)$ | $0$ |
| $4$ | $6$ | $(1,2,4,3,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
