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:
| $r_{ 1 }$ |
$=$ |
$ 36 + 57\cdot 79 + 33\cdot 79^{2} + 40\cdot 79^{3} + 42\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 51 + \left(a + 2\right)\cdot 79 + \left(7 a + 39\right)\cdot 79^{2} + \left(a + 38\right)\cdot 79^{3} + \left(12 a + 52\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 45\cdot 79 + 27\cdot 79^{2} + 50\cdot 79^{3} + 57\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 66 a + 35 + \left(77 a + 27\right)\cdot 79 + \left(34 a + 4\right)\cdot 79^{2} + \left(47 a + 50\right)\cdot 79^{3} + \left(76 a + 13\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 a + 59 + \left(55 a + 70\right)\cdot 79 + \left(73 a + 53\right)\cdot 79^{2} + \left(33 a + 40\right)\cdot 79^{3} + \left(41 a + 57\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 76 a + 54 + 77 a\cdot 79 + \left(71 a + 45\right)\cdot 79^{2} + \left(77 a + 32\right)\cdot 79^{3} + \left(66 a + 63\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 13 a + 22 + \left(a + 39\right)\cdot 79 + \left(44 a + 40\right)\cdot 79^{2} + \left(31 a + 62\right)\cdot 79^{3} + \left(2 a + 42\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 25 a + 34 + \left(23 a + 72\right)\cdot 79 + \left(5 a + 71\right)\cdot 79^{2} + 45 a\cdot 79^{3} + \left(37 a + 65\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,3,2)(4,8,5,7)$ |
| $(1,4,6)(2,3,5)$ |
| $(1,3)(2,4)(5,6)$ |
| $(1,8,3,7)(2,5,6,4)$ |
| $(1,3)(2,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $-4$ |
| $12$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $8$ | $3$ | $(1,4,6)(2,3,5)$ | $1$ |
| $6$ | $4$ | $(1,8,3,7)(2,5,6,4)$ | $0$ |
| $8$ | $6$ | $(1,5,6,3,4,2)(7,8)$ | $-1$ |
| $6$ | $8$ | $(1,2,8,5,3,6,7,4)$ | $0$ |
| $6$ | $8$ | $(1,6,8,4,3,2,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
