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:
| $r_{ 1 }$ |
$=$ |
$ 34 a + 64 + \left(74 a + 61\right)\cdot 97 + \left(70 a + 24\right)\cdot 97^{2} + \left(10 a + 88\right)\cdot 97^{3} + \left(90 a + 69\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 74 + 72\cdot 97 + 88\cdot 97^{2} + 83\cdot 97^{3} + 14\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 88\cdot 97^{2} + 87\cdot 97^{3} + 19\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 88 a + 5 + \left(22 a + 11\right)\cdot 97 + \left(6 a + 91\right)\cdot 97^{2} + \left(70 a + 17\right)\cdot 97^{3} + \left(24 a + 41\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 63 a + 1 + \left(22 a + 5\right)\cdot 97 + \left(26 a + 21\right)\cdot 97^{2} + \left(86 a + 28\right)\cdot 97^{3} + \left(6 a + 52\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 93 + \left(74 a + 42\right)\cdot 97 + \left(90 a + 74\right)\cdot 97^{2} + \left(26 a + 81\right)\cdot 97^{3} + \left(72 a + 92\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3,4,6,5)$ |
| $(1,3)(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-1$ |
| $15$ |
$2$ |
$(1,5)(4,6)$ |
$1$ |
| $20$ |
$3$ |
$(1,3,6)(2,4,5)$ |
$-1$ |
| $30$ |
$4$ |
$(1,6,5,4)$ |
$1$ |
| $24$ |
$5$ |
$(1,4,3,2,6)$ |
$0$ |
| $20$ |
$6$ |
$(1,2,3,4,6,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
