Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 179 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 179 }$: $ x^{2} + 172 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 137 + 43\cdot 179 + 4\cdot 179^{2} + 66\cdot 179^{3} + 59\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 132 a + 166 + \left(92 a + 17\right)\cdot 179 + \left(166 a + 80\right)\cdot 179^{2} + \left(62 a + 103\right)\cdot 179^{3} + \left(150 a + 20\right)\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 162 + 68\cdot 179 + 31\cdot 179^{2} + 3\cdot 179^{3} + 138\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 165\cdot 179 + 138\cdot 179^{2} + 95\cdot 179^{3} + 121\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 a + 16 + \left(86 a + 177\right)\cdot 179 + \left(12 a + 78\right)\cdot 179^{2} + \left(116 a + 19\right)\cdot 179^{3} + \left(28 a + 115\right)\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 a + 92 + \left(117 a + 94\right)\cdot 179 + \left(47 a + 172\right)\cdot 179^{2} + \left(39 a + 10\right)\cdot 179^{3} + \left(122 a + 170\right)\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 128 a + 91 + \left(61 a + 148\right)\cdot 179 + \left(131 a + 30\right)\cdot 179^{2} + \left(139 a + 59\right)\cdot 179^{3} + \left(56 a + 91\right)\cdot 179^{4} +O\left(179^{ 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.
