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:
| $r_{ 1 }$ |
$=$ |
$ 9 + 64\cdot 79 + 9\cdot 79^{2} + 45\cdot 79^{3} + 74\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 a + 45 + \left(33 a + 9\right)\cdot 79 + \left(78 a + 50\right)\cdot 79^{2} + \left(51 a + 8\right)\cdot 79^{3} + \left(41 a + 56\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 a + 13 + \left(45 a + 75\right)\cdot 79 + 15\cdot 79^{2} + \left(27 a + 61\right)\cdot 79^{3} + \left(37 a + 45\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 63 a + 69 + \left(34 a + 3\right)\cdot 79 + \left(26 a + 30\right)\cdot 79^{2} + \left(37 a + 17\right)\cdot 79^{3} + 46\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 50 + 29\cdot 79 + 30\cdot 79^{2} + 76\cdot 79^{3} + 4\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 a + 53 + \left(44 a + 54\right)\cdot 79 + \left(52 a + 21\right)\cdot 79^{2} + \left(41 a + 28\right)\cdot 79^{3} + \left(78 a + 9\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,3)(5,6)$ |
| $(1,3,4,2,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,4)(2,3)(5,6)$ | $-2$ |
| $15$ | $2$ | $(2,4)(5,6)$ | $0$ |
| $20$ | $3$ | $(1,4,5)(2,6,3)$ | $1$ |
| $30$ | $4$ | $(2,5,4,6)$ | $0$ |
| $24$ | $5$ | $(1,3,4,6,2)$ | $-1$ |
| $20$ | $6$ | $(1,3,4,2,5,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
