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:
| $r_{ 1 }$ |
$=$ |
$ 66 a + 83 + \left(42 a + 10\right)\cdot 97 + \left(45 a + 53\right)\cdot 97^{2} + \left(38 a + 30\right)\cdot 97^{3} + \left(58 a + 27\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 + 95\cdot 97 + 93\cdot 97^{2} + 18\cdot 97^{3} + 33\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 a + 52 + \left(54 a + 84\right)\cdot 97 + \left(51 a + 55\right)\cdot 97^{2} + \left(58 a + 23\right)\cdot 97^{3} + \left(38 a + 47\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 a + 83 + \left(23 a + 29\right)\cdot 97 + \left(21 a + 27\right)\cdot 97^{2} + \left(16 a + 56\right)\cdot 97^{3} + \left(85 a + 71\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 78 a + 5 + \left(73 a + 34\right)\cdot 97 + \left(75 a + 25\right)\cdot 97^{2} + \left(80 a + 51\right)\cdot 97^{3} + \left(11 a + 43\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 + 36\cdot 97 + 35\cdot 97^{2} + 13\cdot 97^{3} + 68\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 |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $15$ | $2$ | $(1,2)$ | $-3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $40$ | $3$ | $(1,2,3)$ | $2$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
