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:
| $r_{ 1 }$ |
$=$ |
$ 75 a + 73 + \left(12 a + 4\right)\cdot 79 + \left(39 a + 40\right)\cdot 79^{2} + \left(a + 20\right)\cdot 79^{3} + \left(23 a + 58\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 49 a + 77 + \left(52 a + 12\right)\cdot 79 + \left(26 a + 11\right)\cdot 79^{2} + \left(2 a + 63\right)\cdot 79^{3} + \left(27 a + 23\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 4 a + 69 + \left(66 a + 21\right)\cdot 79 + \left(39 a + 66\right)\cdot 79^{2} + \left(77 a + 61\right)\cdot 79^{3} + 55 a\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 52\cdot 79 + 51\cdot 79^{2} + 75\cdot 79^{3} + 19\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 47 + \left(26 a + 16\right)\cdot 79 + \left(52 a + 64\right)\cdot 79^{2} + \left(76 a + 38\right)\cdot 79^{3} + \left(51 a + 48\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 49\cdot 79 + 3\cdot 79^{2} + 56\cdot 79^{3} + 6\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(2,5)$ |
| $(2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(2,5)$ | $-2$ |
| $9$ | $2$ | $(1,3)(2,5)$ | $0$ |
| $4$ | $3$ | $(1,3,4)(2,5,6)$ | $-2$ |
| $4$ | $3$ | $(1,3,4)$ | $1$ |
| $18$ | $4$ | $(1,2,3,5)(4,6)$ | $0$ |
| $12$ | $6$ | $(1,5,3,6,4,2)$ | $0$ |
| $12$ | $6$ | $(1,3,4)(2,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
