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:
| $r_{ 1 }$ |
$=$ |
$ 23 + 66\cdot 79 + 25\cdot 79^{2} + 79^{3} + 53\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 46 + \left(52 a + 34\right)\cdot 79 + \left(69 a + 4\right)\cdot 79^{2} + \left(16 a + 59\right)\cdot 79^{3} + \left(15 a + 38\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 22\cdot 79 + 76\cdot 79^{2} + 59\cdot 79^{3} + 73\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 a + 29 + \left(48 a + 18\right)\cdot 79 + \left(64 a + 46\right)\cdot 79^{2} + \left(4 a + 45\right)\cdot 79^{3} + \left(28 a + 5\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 49 a + 76 + \left(26 a + 56\right)\cdot 79 + \left(9 a + 21\right)\cdot 79^{2} + \left(62 a + 6\right)\cdot 79^{3} + \left(63 a + 37\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 a + 57 + \left(30 a + 38\right)\cdot 79 + \left(14 a + 62\right)\cdot 79^{2} + \left(74 a + 64\right)\cdot 79^{3} + \left(50 a + 28\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(3,5)$ | $-2$ |
| $9$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,4,6)(2,3,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,5,6,2)$ | $0$ |
| $12$ | $6$ | $(1,4,6)(3,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
