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:
| $r_{ 1 }$ |
$=$ |
$ 36 a + 66 + \left(63 a + 19\right)\cdot 79 + \left(42 a + 42\right)\cdot 79^{2} + \left(8 a + 37\right)\cdot 79^{3} + \left(40 a + 56\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 a + 23 + \left(15 a + 47\right)\cdot 79 + \left(36 a + 21\right)\cdot 79^{2} + \left(70 a + 3\right)\cdot 79^{3} + \left(38 a + 9\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 75\cdot 79 + 28\cdot 79^{2} + 76\cdot 79^{3} + 49\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 44 + \left(29 a + 18\right)\cdot 79 + \left(17 a + 3\right)\cdot 79^{2} + \left(a + 72\right)\cdot 79^{3} + \left(12 a + 4\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 a + 72 + \left(49 a + 19\right)\cdot 79 + \left(61 a + 70\right)\cdot 79^{2} + \left(77 a + 55\right)\cdot 79^{3} + \left(66 a + 15\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 56\cdot 79 + 70\cdot 79^{2} + 70\cdot 79^{3} + 21\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)$ |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,6)$ | $0$ |
| $9$ | $2$ | $(2,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,2,6)$ | $-2$ |
| $4$ | $3$ | $(1,2,6)(3,4,5)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,5,6,3)$ | $-1$ |
| $12$ | $6$ | $(2,6)(3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
