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:
| $r_{ 1 }$ |
$=$ |
$ 2 a + 6 + \left(6 a + 34\right)\cdot 97 + \left(86 a + 36\right)\cdot 97^{2} + \left(11 a + 48\right)\cdot 97^{3} + \left(21 a + 56\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 65 a + 64 + \left(20 a + 78\right)\cdot 97 + \left(46 a + 64\right)\cdot 97^{2} + \left(79 a + 16\right)\cdot 97^{3} + \left(70 a + 49\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 a + 8 + \left(9 a + 83\right)\cdot 97 + \left(49 a + 68\right)\cdot 97^{2} + \left(14 a + 69\right)\cdot 97^{3} + \left(93 a + 48\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 95 a + 8 + \left(90 a + 38\right)\cdot 97 + \left(10 a + 19\right)\cdot 97^{2} + \left(85 a + 71\right)\cdot 97^{3} + \left(75 a + 65\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 32 a + 32 + \left(76 a + 34\right)\cdot 97 + \left(50 a + 90\right)\cdot 97^{2} + \left(17 a + 49\right)\cdot 97^{3} + \left(26 a + 40\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a + 78 + \left(87 a + 22\right)\cdot 97 + \left(47 a + 11\right)\cdot 97^{2} + \left(82 a + 35\right)\cdot 97^{3} + \left(3 a + 30\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $-3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
