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:
| $r_{ 1 }$ |
$=$ |
$ 79 a + 49 + \left(46 a + 18\right)\cdot 97 + \left(80 a + 53\right)\cdot 97^{2} + \left(18 a + 31\right)\cdot 97^{3} + \left(9 a + 27\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 88 + \left(22 a + 69\right)\cdot 97 + \left(68 a + 46\right)\cdot 97^{2} + \left(54 a + 45\right)\cdot 97^{3} + \left(64 a + 30\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 31 + \left(50 a + 83\right)\cdot 97 + \left(16 a + 86\right)\cdot 97^{2} + \left(78 a + 66\right)\cdot 97^{3} + \left(87 a + 17\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 79 + 74\cdot 97 + 30\cdot 97^{2} + 59\cdot 97^{3} + 62\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 36 a + 41 + \left(69 a + 72\right)\cdot 97 + \left(3 a + 71\right)\cdot 97^{2} + \left(75 a + 40\right)\cdot 97^{3} + \left(26 a + 80\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 a + 23 + \left(74 a + 60\right)\cdot 97 + \left(28 a + 92\right)\cdot 97^{2} + \left(42 a + 31\right)\cdot 97^{3} + \left(32 a + 40\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 61 a + 77 + \left(27 a + 8\right)\cdot 97 + \left(93 a + 6\right)\cdot 97^{2} + \left(21 a + 15\right)\cdot 97^{3} + \left(70 a + 32\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 7 }$
| Cycle notation |
| $(1,2,3,4,5,6,7)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 7 }$
| Character value |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $-4$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $-1$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $210$ | $4$ | $(1,2,3,4)$ | $2$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $-1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
