Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$: $ x^{2} + 96 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 a + \left(55 a + 63\right)\cdot 97 + \left(54 a + 50\right)\cdot 97^{2} + \left(21 a + 94\right)\cdot 97^{3} + \left(57 a + 76\right)\cdot 97^{4} + \left(40 a + 88\right)\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 a + 30 + \left(41 a + 88\right)\cdot 97 + \left(42 a + 49\right)\cdot 97^{2} + \left(75 a + 61\right)\cdot 97^{3} + \left(39 a + 15\right)\cdot 97^{4} + \left(56 a + 72\right)\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 68 + \left(55 a + 8\right)\cdot 97 + \left(54 a + 47\right)\cdot 97^{2} + \left(21 a + 35\right)\cdot 97^{3} + \left(57 a + 81\right)\cdot 97^{4} + \left(40 a + 24\right)\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 67 a + 1 + \left(41 a + 34\right)\cdot 97 + \left(42 a + 46\right)\cdot 97^{2} + \left(75 a + 2\right)\cdot 97^{3} + \left(39 a + 20\right)\cdot 97^{4} + \left(56 a + 8\right)\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 81 + 72\cdot 97 + 79\cdot 97^{2} + 64\cdot 97^{3} + 83\cdot 97^{4} + 6\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 24\cdot 97 + 17\cdot 97^{2} + 32\cdot 97^{3} + 13\cdot 97^{4} + 90\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,2)(3,4,6)$ |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $3$ | $2$ | $(2,3)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $1$ |
| $8$ | $3$ | $(1,5,2)(3,4,6)$ | $0$ |
| $6$ | $4$ | $(1,3,4,2)$ | $-1$ |
| $6$ | $4$ | $(1,4)(2,6,3,5)$ | $1$ |
| $8$ | $6$ | $(1,5,2,4,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
