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:
| $r_{ 1 }$ |
$=$ |
$ 62 a + 71 + \left(13 a + 53\right)\cdot 97 + \left(63 a + 5\right)\cdot 97^{2} + \left(27 a + 78\right)\cdot 97^{3} + \left(25 a + 1\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 63 + 17\cdot 97 + 61\cdot 97^{2} + 16\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 22 a + 55 + \left(38 a + 31\right)\cdot 97 + \left(79 a + 94\right)\cdot 97^{2} + \left(23 a + 67\right)\cdot 97^{3} + \left(82 a + 29\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 36 + \left(83 a + 5\right)\cdot 97 + \left(33 a + 55\right)\cdot 97^{2} + \left(69 a + 42\right)\cdot 97^{3} + \left(71 a + 96\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 a + 77 + \left(58 a + 47\right)\cdot 97 + \left(17 a + 38\right)\cdot 97^{2} + \left(73 a + 12\right)\cdot 97^{3} + \left(14 a + 88\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 88 + 37\cdot 97 + 36\cdot 97^{2} + 73\cdot 97^{3} + 95\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $-2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
