Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 26 + \left(75 a + 15\right)\cdot 79 + \left(36 a + 19\right)\cdot 79^{2} + \left(10 a + 13\right)\cdot 79^{3} + \left(26 a + 71\right)\cdot 79^{4} + \left(45 a + 29\right)\cdot 79^{5} + \left(42 a + 1\right)\cdot 79^{6} + \left(22 a + 10\right)\cdot 79^{7} + \left(46 a + 67\right)\cdot 79^{8} + \left(64 a + 69\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 a + 50 + \left(38 a + 9\right)\cdot 79 + \left(22 a + 8\right)\cdot 79^{2} + \left(71 a + 15\right)\cdot 79^{3} + \left(9 a + 70\right)\cdot 79^{4} + \left(16 a + 75\right)\cdot 79^{5} + \left(39 a + 27\right)\cdot 79^{6} + \left(8 a + 15\right)\cdot 79^{7} + \left(71 a + 8\right)\cdot 79^{8} + \left(38 a + 16\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 68\cdot 79 + 71\cdot 79^{2} + 28\cdot 79^{3} + 28\cdot 79^{4} + 15\cdot 79^{5} + 75\cdot 79^{6} + 57\cdot 79^{7} + 6\cdot 79^{8} + 68\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 a + 53 + \left(3 a + 63\right)\cdot 79 + \left(42 a + 59\right)\cdot 79^{2} + \left(68 a + 65\right)\cdot 79^{3} + \left(52 a + 7\right)\cdot 79^{4} + \left(33 a + 49\right)\cdot 79^{5} + \left(36 a + 77\right)\cdot 79^{6} + \left(56 a + 68\right)\cdot 79^{7} + \left(32 a + 11\right)\cdot 79^{8} + \left(14 a + 9\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 29 + \left(40 a + 69\right)\cdot 79 + \left(56 a + 70\right)\cdot 79^{2} + \left(7 a + 63\right)\cdot 79^{3} + \left(69 a + 8\right)\cdot 79^{4} + \left(62 a + 3\right)\cdot 79^{5} + \left(39 a + 51\right)\cdot 79^{6} + \left(70 a + 63\right)\cdot 79^{7} + \left(7 a + 70\right)\cdot 79^{8} + \left(40 a + 62\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 + 10\cdot 79 + 7\cdot 79^{2} + 50\cdot 79^{3} + 50\cdot 79^{4} + 63\cdot 79^{5} + 3\cdot 79^{6} + 21\cdot 79^{7} + 72\cdot 79^{8} + 10\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)$ |
| $(2,5)$ |
| $(1,3,2)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(2,5)$ | $1$ |
| $3$ | $2$ | $(1,4)(2,5)$ | $-1$ |
| $4$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $4$ | $6$ | $(1,3,2,4,6,5)$ | $0$ |
| $4$ | $6$ | $(1,5,6,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
