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:
| $r_{ 1 }$ |
$=$ |
$ 31 a + 56 + \left(5 a + 2\right)\cdot 79 + \left(43 a + 50\right)\cdot 79^{2} + \left(20 a + 11\right)\cdot 79^{3} + \left(58 a + 74\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 a + 8 + \left(73 a + 56\right)\cdot 79 + \left(35 a + 8\right)\cdot 79^{2} + \left(58 a + 68\right)\cdot 79^{3} + \left(20 a + 32\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 a + 73 + \left(21 a + 68\right)\cdot 79 + \left(54 a + 8\right)\cdot 79^{2} + \left(41 a + 12\right)\cdot 79^{3} + \left(57 a + 73\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 41\cdot 79 + 67\cdot 79^{2} + 19\cdot 79^{3} + 75\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 a + 47 + \left(57 a + 37\right)\cdot 79 + \left(24 a + 41\right)\cdot 79^{2} + \left(37 a + 78\right)\cdot 79^{3} + \left(21 a + 9\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 62 + 29\cdot 79 + 60\cdot 79^{2} + 46\cdot 79^{3} + 50\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,4)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $0$ |
| $6$ | $2$ | $(2,4)$ | $-2$ |
| $9$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,4)$ | $1$ |
| $4$ | $3$ | $(1,2,4)(3,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,4,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,4,3)$ | $0$ |
| $12$ | $6$ | $(2,4)(3,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
