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:
| $r_{ 1 }$ |
$=$ |
$ 49 a + 17 + \left(10 a + 52\right)\cdot 79 + \left(25 a + 61\right)\cdot 79^{2} + \left(67 a + 7\right)\cdot 79^{3} + \left(43 a + 45\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 70 + 68\cdot 79 + 32\cdot 79^{2} + 21\cdot 79^{3} + 61\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 a + 68 + \left(44 a + 8\right)\cdot 79 + \left(17 a + 45\right)\cdot 79^{2} + \left(51 a + 40\right)\cdot 79^{3} + \left(50 a + 46\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 75 + 71\cdot 79 + 2\cdot 79^{2} + 43\cdot 79^{3} + 16\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 a + 66 + \left(68 a + 13\right)\cdot 79 + \left(53 a + 76\right)\cdot 79^{2} + \left(11 a + 49\right)\cdot 79^{3} + \left(35 a + 21\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 a + 21 + \left(34 a + 21\right)\cdot 79 + \left(61 a + 18\right)\cdot 79^{2} + \left(27 a + 74\right)\cdot 79^{3} + \left(28 a + 45\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3,5,6,4)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,5)$ | $1$ |
| $20$ | $3$ | $(1,3,6)(2,5,4)$ | $-1$ |
| $30$ | $4$ | $(1,5,2,3)$ | $-1$ |
| $24$ | $5$ | $(2,4,3,5,6)$ | $0$ |
| $20$ | $6$ | $(1,2,3,5,6,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
