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:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 35 + \left(70 a + 5\right)\cdot 79 + \left(28 a + 33\right)\cdot 79^{2} + \left(44 a + 39\right)\cdot 79^{3} + \left(41 a + 13\right)\cdot 79^{4} + \left(27 a + 73\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 67 a + 47 + \left(8 a + 63\right)\cdot 79 + \left(50 a + 70\right)\cdot 79^{2} + \left(34 a + 54\right)\cdot 79^{3} + \left(37 a + 10\right)\cdot 79^{4} + \left(51 a + 59\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 75 a + 69 + \left(53 a + 75\right)\cdot 79 + \left(76 a + 29\right)\cdot 79^{2} + \left(53 a + 46\right)\cdot 79^{3} + \left(11 a + 23\right)\cdot 79^{4} + \left(31 a + 71\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 77 + 9\cdot 79 + 54\cdot 79^{2} + 63\cdot 79^{3} + 54\cdot 79^{4} + 25\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 27\cdot 79 + 75\cdot 79^{2} + 8\cdot 79^{3} + 74\cdot 79^{4} + 74\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 65 + \left(25 a + 54\right)\cdot 79 + \left(2 a + 52\right)\cdot 79^{2} + \left(25 a + 23\right)\cdot 79^{3} + \left(67 a + 60\right)\cdot 79^{4} + \left(47 a + 11\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,6)$ |
| $(1,3)(2,6)(4,5)$ |
| $(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $0$ |
| $3$ | $2$ | $(1,6)(2,3)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,4,2)(3,5,6)$ | $-1$ |
| $2$ | $6$ | $(1,5,2,3,4,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
