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:
| $r_{ 1 }$ |
$=$ |
$ 48 + 65\cdot 79 + 74\cdot 79^{2} + 23\cdot 79^{3} + 24\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 a + 75 + \left(34 a + 44\right)\cdot 79 + 22\cdot 79^{2} + \left(70 a + 42\right)\cdot 79^{3} + \left(8 a + 61\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 a + 25 + \left(24 a + 45\right)\cdot 79 + 8\cdot 79^{2} + \left(5 a + 27\right)\cdot 79^{3} + \left(53 a + 51\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 a + 23 + \left(44 a + 52\right)\cdot 79 + \left(78 a + 67\right)\cdot 79^{2} + \left(8 a + 32\right)\cdot 79^{3} + 70 a\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 66 + \left(54 a + 28\right)\cdot 79 + \left(78 a + 63\right)\cdot 79^{2} + \left(73 a + 31\right)\cdot 79^{3} + \left(25 a + 20\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
