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:
| $r_{ 1 }$ |
$=$ |
$ 92 a + 95 + \left(58 a + 84\right)\cdot 97 + \left(13 a + 51\right)\cdot 97^{2} + \left(76 a + 69\right)\cdot 97^{3} + \left(25 a + 42\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 77 + 83\cdot 97 + 67\cdot 97^{2} + 62\cdot 97^{3} + 42\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 90 + \left(38 a + 51\right)\cdot 97 + \left(83 a + 6\right)\cdot 97^{2} + \left(20 a + 35\right)\cdot 97^{3} + \left(71 a + 89\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 45 + \left(4 a + 3\right)\cdot 97 + \left(20 a + 26\right)\cdot 97^{2} + \left(49 a + 47\right)\cdot 97^{3} + 82\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 a + 83 + \left(92 a + 66\right)\cdot 97 + \left(76 a + 41\right)\cdot 97^{2} + \left(47 a + 76\right)\cdot 97^{3} + \left(96 a + 33\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
