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:
| $r_{ 1 }$ |
$=$ |
$ 77 a + 10 + \left(85 a + 44\right)\cdot 97 + \left(22 a + 31\right)\cdot 97^{2} + \left(24 a + 96\right)\cdot 97^{3} + \left(83 a + 18\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 82 a + 56 + \left(76 a + 2\right)\cdot 97 + \left(81 a + 46\right)\cdot 97^{2} + \left(43 a + 67\right)\cdot 97^{3} + \left(10 a + 16\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 a + 32 + \left(90 a + 68\right)\cdot 97 + \left(60 a + 14\right)\cdot 97^{2} + \left(13 a + 72\right)\cdot 97^{3} + \left(78 a + 64\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 a + 87 + \left(11 a + 52\right)\cdot 97 + \left(74 a + 65\right)\cdot 97^{2} + 72 a\cdot 97^{3} + \left(13 a + 78\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + 41 + \left(20 a + 94\right)\cdot 97 + \left(15 a + 50\right)\cdot 97^{2} + \left(53 a + 29\right)\cdot 97^{3} + \left(86 a + 80\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 a + 65 + \left(6 a + 28\right)\cdot 97 + \left(36 a + 82\right)\cdot 97^{2} + \left(83 a + 24\right)\cdot 97^{3} + \left(18 a + 32\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,6,4,2,3)$ |
| $(1,4)(2,5)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
