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:
| $r_{ 1 }$ |
$=$ |
$ 78 a + 40 + \left(3 a + 12\right)\cdot 97 + \left(54 a + 72\right)\cdot 97^{2} + \left(94 a + 19\right)\cdot 97^{3} + \left(95 a + 89\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 + 27\cdot 97 + 78\cdot 97^{2} + 35\cdot 97^{3} + 8\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 14 + \left(20 a + 56\right)\cdot 97 + \left(76 a + 62\right)\cdot 97^{2} + \left(5 a + 12\right)\cdot 97^{3} + \left(28 a + 34\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 72 a + 28 + \left(39 a + 22\right)\cdot 97 + \left(22 a + 24\right)\cdot 97^{2} + \left(13 a + 66\right)\cdot 97^{3} + \left(61 a + 30\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 a + 21 + \left(93 a + 35\right)\cdot 97 + \left(42 a + 25\right)\cdot 97^{2} + \left(2 a + 60\right)\cdot 97^{3} + \left(a + 90\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 3 + \left(57 a + 87\right)\cdot 97 + \left(74 a + 6\right)\cdot 97^{2} + \left(83 a + 57\right)\cdot 97^{3} + \left(35 a + 78\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 71 a + 40 + \left(76 a + 50\right)\cdot 97 + \left(20 a + 21\right)\cdot 97^{2} + \left(91 a + 39\right)\cdot 97^{3} + \left(68 a + 56\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 |
| $1$ | $1$ | $()$ | $21$ |
| $21$ | $2$ | $(1,2)$ | $-1$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $70$ | $3$ | $(1,2,3)$ | $-3$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $210$ | $4$ | $(1,2,3,4)$ | $1$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $1$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $-1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
