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:
| $r_{ 1 }$ |
$=$ |
$ 57 a + 72 + \left(55 a + 34\right)\cdot 79 + \left(12 a + 69\right)\cdot 79^{2} + \left(59 a + 61\right)\cdot 79^{3} + \left(75 a + 16\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 8 + \left(16 a + 37\right)\cdot 79 + \left(41 a + 6\right)\cdot 79^{2} + \left(38 a + 13\right)\cdot 79^{3} + \left(48 a + 47\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 10\cdot 79 + 62\cdot 79^{2} + 66\cdot 79^{3} + 28\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 18\cdot 79 + 41\cdot 79^{2} + 55\cdot 79^{3} + 53\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a + 50 + \left(23 a + 33\right)\cdot 79 + \left(66 a + 26\right)\cdot 79^{2} + \left(19 a + 29\right)\cdot 79^{3} + \left(3 a + 33\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 a + 38 + \left(62 a + 23\right)\cdot 79 + \left(37 a + 31\right)\cdot 79^{2} + \left(40 a + 10\right)\cdot 79^{3} + \left(30 a + 57\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,2)(3,4)(5,6)$ |
| $(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(2,4)$ |
$2$ |
| $9$ |
$2$ |
$(1,3)(2,4)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$-2$ |
| $4$ |
$3$ |
$(1,3,5)$ |
$1$ |
| $18$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,3,6,5,2)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,5)(2,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
