Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 47\cdot 79 + 57\cdot 79^{2} + 67\cdot 79^{3} + 72\cdot 79^{4} + 5\cdot 79^{5} + 58\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 a + 13 + \left(5 a + 19\right)\cdot 79 + \left(75 a + 61\right)\cdot 79^{2} + \left(36 a + 34\right)\cdot 79^{3} + 59\cdot 79^{4} + \left(54 a + 3\right)\cdot 79^{5} + \left(12 a + 46\right)\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 71 a + 21 + \left(73 a + 16\right)\cdot 79 + \left(3 a + 52\right)\cdot 79^{2} + \left(42 a + 75\right)\cdot 79^{3} + \left(78 a + 22\right)\cdot 79^{4} + \left(24 a + 57\right)\cdot 79^{5} + \left(66 a + 4\right)\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 a + 23 + \left(28 a + 48\right)\cdot 79 + \left(68 a + 66\right)\cdot 79^{2} + \left(29 a + 75\right)\cdot 79^{3} + \left(33 a + 76\right)\cdot 79^{4} + \left(56 a + 78\right)\cdot 79^{5} + \left(68 a + 56\right)\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 68 + \left(50 a + 31\right)\cdot 79 + \left(10 a + 27\right)\cdot 79^{2} + \left(49 a + 37\right)\cdot 79^{3} + \left(45 a + 1\right)\cdot 79^{4} + \left(22 a + 23\right)\cdot 79^{5} + \left(10 a + 69\right)\cdot 79^{6} +O\left(79^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 + 74\cdot 79 + 50\cdot 79^{2} + 24\cdot 79^{3} + 3\cdot 79^{4} + 68\cdot 79^{5} + 79^{6} +O\left(79^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $45$ |
$2$ |
$(1,2)(3,4)$ |
$1$ |
| $40$ |
$3$ |
$(1,2,3)(4,5,6)$ |
$2$ |
| $40$ |
$3$ |
$(1,2,3)$ |
$-1$ |
| $90$ |
$4$ |
$(1,2,3,4)(5,6)$ |
$-1$ |
| $72$ |
$5$ |
$(1,2,3,4,5)$ |
$0$ |
| $72$ |
$5$ |
$(1,3,4,5,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
