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:
| $r_{ 1 }$ |
$=$ |
$ 42 a + 29 + \left(5 a + 10\right)\cdot 97 + \left(37 a + 49\right)\cdot 97^{2} + \left(77 a + 79\right)\cdot 97^{3} + \left(52 a + 66\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 + 39\cdot 97 + 27\cdot 97^{2} + 33\cdot 97^{3} + 36\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 10 + \left(80 a + 84\right)\cdot 97 + \left(63 a + 36\right)\cdot 97^{2} + \left(66 a + 71\right)\cdot 97^{3} + \left(87 a + 64\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 a + 71 + \left(91 a + 70\right)\cdot 97 + \left(59 a + 80\right)\cdot 97^{2} + \left(19 a + 22\right)\cdot 97^{3} + \left(44 a + 42\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 84 + 48\cdot 97 + 76\cdot 97^{2} + 9\cdot 97^{3} + 92\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 67 a + 40 + \left(16 a + 37\right)\cdot 97 + \left(33 a + 20\right)\cdot 97^{2} + \left(30 a + 74\right)\cdot 97^{3} + \left(9 a + 85\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,5)(3,4)$ |
| $(1,3,4,5,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,6)(2,5)(3,4)$ | $2$ |
| $15$ | $2$ | $(1,4)(3,6)$ | $0$ |
| $20$ | $3$ | $(1,6,4)(2,5,3)$ | $1$ |
| $30$ | $4$ | $(1,5,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,6,3,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,6,5,4,3)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
