Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 46 + 13\cdot 79 + 62\cdot 79^{2} + 67\cdot 79^{3} + 53\cdot 79^{4} + 28\cdot 79^{5} + 7\cdot 79^{6} + 56\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 44\cdot 79 + 78\cdot 79^{2} + 16\cdot 79^{3} + 65\cdot 79^{4} + 47\cdot 79^{5} + 70\cdot 79^{6} + 40\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 a + 56 + \left(66 a + 15\right)\cdot 79 + \left(33 a + 54\right)\cdot 79^{2} + \left(56 a + 7\right)\cdot 79^{3} + \left(44 a + 49\right)\cdot 79^{4} + \left(78 a + 6\right)\cdot 79^{5} + \left(24 a + 50\right)\cdot 79^{6} + \left(45 a + 57\right)\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 73 + \left(a + 78\right)\cdot 79 + \left(61 a + 19\right)\cdot 79^{2} + \left(60 a + 57\right)\cdot 79^{3} + \left(40 a + 25\right)\cdot 79^{4} + \left(14 a + 30\right)\cdot 79^{5} + \left(17 a + 15\right)\cdot 79^{6} + \left(6 a + 47\right)\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 a + 19 + \left(77 a + 55\right)\cdot 79 + 17 a\cdot 79^{2} + \left(18 a + 57\right)\cdot 79^{3} + \left(38 a + 5\right)\cdot 79^{4} + \left(64 a + 4\right)\cdot 79^{5} + \left(61 a + 18\right)\cdot 79^{6} + \left(72 a + 36\right)\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 30 + \left(12 a + 29\right)\cdot 79 + \left(45 a + 21\right)\cdot 79^{2} + \left(22 a + 30\right)\cdot 79^{3} + \left(34 a + 37\right)\cdot 79^{4} + 40\cdot 79^{5} + \left(54 a + 75\right)\cdot 79^{6} + \left(33 a + 77\right)\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(3,5)(4,6)$ |
| $(1,3,5)(2,4,6)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $1$ |
| $6$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(3,6,4,5)$ | $1$ |
| $6$ | $4$ | $(1,6,2,5)(3,4)$ | $-1$ |
| $8$ | $6$ | $(1,3,6,2,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
