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:
| $r_{ 1 }$ |
$=$ |
$ 88 a + 20 + \left(25 a + 13\right)\cdot 97 + \left(78 a + 5\right)\cdot 97^{2} + \left(a + 19\right)\cdot 97^{3} + \left(67 a + 6\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 11 + \left(71 a + 48\right)\cdot 97 + \left(18 a + 57\right)\cdot 97^{2} + \left(95 a + 39\right)\cdot 97^{3} + \left(29 a + 71\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 42 a + 59 + \left(58 a + 84\right)\cdot 97 + \left(48 a + 94\right)\cdot 97^{2} + \left(34 a + 6\right)\cdot 97^{3} + \left(83 a + 25\right)\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 76 + 97 + 93\cdot 97^{2} + 17\cdot 97^{3} + 18\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 14 + 17\cdot 97 + 17\cdot 97^{2} + 28\cdot 97^{3} + 74\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 + 25\cdot 97 + 35\cdot 97^{2} + 89\cdot 97^{3} + 21\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 a + 4 + \left(38 a + 4\right)\cdot 97 + \left(48 a + 85\right)\cdot 97^{2} + \left(62 a + 89\right)\cdot 97^{3} + \left(13 a + 73\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 |
| $1$ | $1$ | $()$ | $14$ |
| $21$ | $2$ | $(1,2)$ | $6$ |
| $105$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $105$ | $2$ | $(1,2)(3,4)$ | $2$ |
| $70$ | $3$ | $(1,2,3)$ | $2$ |
| $280$ | $3$ | $(1,2,3)(4,5,6)$ | $-1$ |
| $210$ | $4$ | $(1,2,3,4)$ | $0$ |
| $630$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $504$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $210$ | $6$ | $(1,2,3)(4,5)(6,7)$ | $2$ |
| $420$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
| $840$ | $6$ | $(1,2,3,4,5,6)$ | $-1$ |
| $720$ | $7$ | $(1,2,3,4,5,6,7)$ | $0$ |
| $504$ | $10$ | $(1,2,3,4,5)(6,7)$ | $1$ |
| $420$ | $12$ | $(1,2,3,4)(5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
