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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 48 + \left(10 a + 2\right)\cdot 97 + \left(39 a + 73\right)\cdot 97^{2} + \left(81 a + 51\right)\cdot 97^{3} + \left(61 a + 56\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 78 + \left(92 a + 9\right)\cdot 97 + \left(53 a + 78\right)\cdot 97^{2} + \left(53 a + 53\right)\cdot 97^{3} + \left(81 a + 70\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 86 + 50\cdot 97 + 71\cdot 97^{2} + 31\cdot 97^{3} + 27\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 90 a + 55 + \left(86 a + 5\right)\cdot 97 + \left(57 a + 5\right)\cdot 97^{2} + \left(15 a + 94\right)\cdot 97^{3} + \left(35 a + 36\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 91 a + 84 + \left(4 a + 95\right)\cdot 97 + \left(43 a + 39\right)\cdot 97^{2} + \left(43 a + 53\right)\cdot 97^{3} + \left(15 a + 1\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 29\cdot 97 + 23\cdot 97^{2} + 6\cdot 97^{3} + 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,6)$ |
| $(1,2)(3,4)(5,6)$ |
| $(1,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(3,5)$ | $0$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $1$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
