Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 58 + 50\cdot 79 + 70\cdot 79^{2} + 2\cdot 79^{3} + 14\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 7 + \left(75 a + 8\right)\cdot 79 + \left(12 a + 5\right)\cdot 79^{2} + \left(11 a + 21\right)\cdot 79^{3} + \left(40 a + 9\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 43 + \left(72 a + 25\right)\cdot 79 + \left(9 a + 28\right)\cdot 79^{2} + \left(55 a + 22\right)\cdot 79^{3} + \left(65 a + 46\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 a + 73 + \left(6 a + 67\right)\cdot 79 + \left(69 a + 44\right)\cdot 79^{2} + \left(23 a + 67\right)\cdot 79^{3} + \left(13 a + 56\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 72 a + 14 + \left(3 a + 76\right)\cdot 79 + \left(66 a + 21\right)\cdot 79^{2} + \left(67 a + 19\right)\cdot 79^{3} + \left(38 a + 38\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 8\cdot 79 + 66\cdot 79^{2} + 24\cdot 79^{3} + 72\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,6,5,3,2)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(4,6)$ |
$0$ |
| $20$ |
$3$ |
$(1,6,3)(2,4,5)$ |
$1$ |
| $30$ |
$4$ |
$(1,4,2,6)$ |
$0$ |
| $24$ |
$5$ |
$(1,3,4,6,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,4,6,5,3,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
