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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$15$ |
| $21$ |
$2$ |
$(1,2)$ |
$5$ |
| $105$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $105$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $70$ |
$3$ |
$(1,2,3)$ |
$3$ |
| $280$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$0$ |
| $210$ |
$4$ |
$(1,2,3,4)$ |
$1$ |
| $630$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $504$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $210$ |
$6$ |
$(1,2,3)(4,5)(6,7)$ |
$-1$ |
| $420$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
| $840$ |
$6$ |
$(1,2,3,4,5,6)$ |
$0$ |
| $720$ |
$7$ |
$(1,2,3,4,5,6,7)$ |
$1$ |
| $504$ |
$10$ |
$(1,2,3,4,5)(6,7)$ |
$0$ |
| $420$ |
$12$ |
$(1,2,3,4)(5,6,7)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
