Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 64\cdot 79 + 50\cdot 79^{2} + 37\cdot 79^{3} + 77\cdot 79^{4} + 58\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 8 + \left(74 a + 74\right)\cdot 79 + \left(11 a + 33\right)\cdot 79^{2} + \left(3 a + 25\right)\cdot 79^{3} + \left(20 a + 18\right)\cdot 79^{4} + \left(20 a + 33\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 a + 35 + \left(56 a + 34\right)\cdot 79 + \left(69 a + 10\right)\cdot 79^{2} + \left(40 a + 25\right)\cdot 79^{3} + \left(a + 55\right)\cdot 79^{4} + \left(78 a + 18\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 + 17\cdot 79 + 57\cdot 79^{2} + 61\cdot 79^{3} + 43\cdot 79^{4} + 77\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 a + 45 + 44\cdot 79 + \left(51 a + 19\right)\cdot 79^{2} + \left(43 a + 1\right)\cdot 79^{3} + \left(7 a + 53\right)\cdot 79^{4} + \left(49 a + 57\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a + 29 + \left(4 a + 48\right)\cdot 79 + \left(67 a + 50\right)\cdot 79^{2} + \left(75 a + 16\right)\cdot 79^{3} + \left(58 a + 35\right)\cdot 79^{4} + \left(58 a + 33\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 22 a + 23 + \left(78 a + 67\right)\cdot 79 + \left(27 a + 69\right)\cdot 79^{2} + \left(35 a + 72\right)\cdot 79^{3} + \left(71 a + 16\right)\cdot 79^{4} + \left(29 a + 20\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 32 a + 3 + \left(22 a + 44\right)\cdot 79 + \left(9 a + 23\right)\cdot 79^{2} + \left(38 a + 75\right)\cdot 79^{3} + \left(77 a + 15\right)\cdot 79^{4} + 16\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,6)(5,7)$ |
| $(1,3,4,6)(2,5,8,7)$ |
| $(2,3,7)(5,8,6)$ |
| $(1,2,4,8)(3,7,6,5)$ |
| $(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,8)(3,6)(5,7)$ |
$-2$ |
$-2$ |
| $12$ |
$2$ |
$(2,6)(3,8)(5,7)$ |
$0$ |
$0$ |
| $8$ |
$3$ |
$(1,3,5)(4,6,7)$ |
$-1$ |
$-1$ |
| $6$ |
$4$ |
$(1,3,4,6)(2,5,8,7)$ |
$0$ |
$0$ |
| $8$ |
$6$ |
$(1,7,3,4,5,6)(2,8)$ |
$1$ |
$1$ |
| $6$ |
$8$ |
$(1,3,7,8,4,6,5,2)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $6$ |
$8$ |
$(1,6,7,2,4,3,5,8)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
