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:
| $r_{ 1 }$ |
$=$ |
$ 44 a + 41 + \left(6 a + 79\right)\cdot 97 + \left(50 a + 92\right)\cdot 97^{2} + \left(65 a + 64\right)\cdot 97^{3} + \left(29 a + 56\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 a + 6 + \left(79 a + 20\right)\cdot 97 + \left(28 a + 74\right)\cdot 97^{2} + \left(39 a + 64\right)\cdot 97^{3} + \left(53 a + 19\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 a + 85 + \left(90 a + 41\right)\cdot 97 + \left(46 a + 39\right)\cdot 97^{2} + \left(31 a + 80\right)\cdot 97^{3} + \left(67 a + 20\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 89\cdot 97 + 25\cdot 97^{2} + 84\cdot 97^{3} + 52\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 59 + 6\cdot 97 + 35\cdot 97^{2} + 18\cdot 97^{3} + 10\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 a + 52 + \left(17 a + 53\right)\cdot 97 + \left(68 a + 23\right)\cdot 97^{2} + \left(57 a + 75\right)\cdot 97^{3} + \left(43 a + 33\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,3,4,5,6,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $15$ | $2$ | $(1,2)(5,6)$ | $1$ |
| $20$ | $3$ | $(1,4,6)(2,3,5)$ | $-1$ |
| $30$ | $4$ | $(1,6,2,5)$ | $1$ |
| $24$ | $5$ | $(1,6,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,3,4,5,6,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
