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:
| $r_{ 1 }$ |
$=$ |
$ 67 a + 5 + \left(41 a + 60\right)\cdot 97 + \left(80 a + 34\right)\cdot 97^{2} + \left(12 a + 48\right)\cdot 97^{3} + 51 a\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 a + 44 + \left(39 a + 3\right)\cdot 97 + \left(6 a + 81\right)\cdot 97^{2} + \left(2 a + 55\right)\cdot 97^{3} + \left(18 a + 32\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 74 + 72\cdot 97 + 36\cdot 97^{2} + 12\cdot 97^{3} + 55\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 a + 72 + \left(55 a + 34\right)\cdot 97 + \left(16 a + 73\right)\cdot 97^{2} + \left(84 a + 77\right)\cdot 97^{3} + \left(45 a + 38\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 85 a + 56 + \left(57 a + 30\right)\cdot 97 + \left(90 a + 48\right)\cdot 97^{2} + \left(94 a + 51\right)\cdot 97^{3} + \left(78 a + 48\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 42 + 89\cdot 97 + 16\cdot 97^{2} + 45\cdot 97^{3} + 18\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$9$ |
| $15$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$3$ |
| $15$ |
$2$ |
$(1,2)$ |
$3$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$1$ |
| $90$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
| $144$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $120$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $120$ |
$6$ |
$(1,2,3)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
