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:
| $r_{ 1 }$ |
$=$ |
$ 70 a + 77 + \left(17 a + 58\right)\cdot 79 + \left(31 a + 63\right)\cdot 79^{2} + \left(a + 15\right)\cdot 79^{3} + \left(36 a + 54\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 68 + \left(61 a + 6\right)\cdot 79 + \left(47 a + 77\right)\cdot 79^{2} + \left(77 a + 64\right)\cdot 79^{3} + \left(42 a + 9\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 70 + 52\cdot 79 + 71\cdot 79^{2} + 77\cdot 79^{3} + 49\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 49 + \left(29 a + 64\right)\cdot 79 + \left(a + 10\right)\cdot 79^{2} + \left(40 a + 50\right)\cdot 79^{3} + \left(60 a + 66\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 62 + 59\cdot 79 + 30\cdot 79^{2} + 18\cdot 79^{3} + 48\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 a + 70 + \left(49 a + 72\right)\cdot 79 + \left(77 a + 61\right)\cdot 79^{2} + \left(38 a + 9\right)\cdot 79^{3} + \left(18 a + 8\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,2,4,5,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $15$ | $2$ | $(1,4)(2,5)$ | $1$ |
| $20$ | $3$ | $(1,5,2)(3,6,4)$ | $-1$ |
| $30$ | $4$ | $(2,6,4,5)$ | $-1$ |
| $24$ | $5$ | $(1,4,6,2,3)$ | $0$ |
| $20$ | $6$ | $(1,6,5,4,2,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
