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:
| $r_{ 1 }$ |
$=$ |
$ 29 + 44\cdot 79 + 16\cdot 79^{2} + 16\cdot 79^{3} + 25\cdot 79^{4} + 20\cdot 79^{5} + 66\cdot 79^{6} + 16\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 + 58\cdot 79 + 36\cdot 79^{2} + 48\cdot 79^{3} + 30\cdot 79^{4} + 25\cdot 79^{5} + 8\cdot 79^{6} + 15\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 9 + \left(60 a + 42\right)\cdot 79 + \left(42 a + 51\right)\cdot 79^{2} + \left(16 a + 66\right)\cdot 79^{3} + \left(45 a + 58\right)\cdot 79^{4} + \left(14 a + 67\right)\cdot 79^{5} + \left(2 a + 52\right)\cdot 79^{6} + \left(a + 42\right)\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 23 + \left(23 a + 72\right)\cdot 79 + \left(46 a + 76\right)\cdot 79^{2} + \left(15 a + 47\right)\cdot 79^{3} + \left(69 a + 69\right)\cdot 79^{4} + \left(76 a + 78\right)\cdot 79^{5} + \left(40 a + 12\right)\cdot 79^{6} + \left(66 a + 8\right)\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 70 a + 32 + \left(55 a + 7\right)\cdot 79 + \left(32 a + 21\right)\cdot 79^{2} + \left(63 a + 17\right)\cdot 79^{3} + \left(9 a + 44\right)\cdot 79^{4} + \left(2 a + 7\right)\cdot 79^{5} + \left(38 a + 56\right)\cdot 79^{6} + \left(12 a + 33\right)\cdot 79^{7} +O\left(79^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 a + 20 + \left(18 a + 12\right)\cdot 79 + \left(36 a + 34\right)\cdot 79^{2} + \left(62 a + 40\right)\cdot 79^{3} + \left(33 a + 8\right)\cdot 79^{4} + \left(64 a + 37\right)\cdot 79^{5} + \left(76 a + 40\right)\cdot 79^{6} + \left(77 a + 41\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$-3$ |
| $3$ |
$2$ |
$(3,4)$ |
$1$ |
| $3$ |
$2$ |
$(3,4)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,5)(2,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-1$ |
| $8$ |
$3$ |
$(1,3,5)(2,4,6)$ |
$0$ |
| $6$ |
$4$ |
$(3,6,4,5)$ |
$1$ |
| $6$ |
$4$ |
$(1,6,2,5)(3,4)$ |
$-1$ |
| $8$ |
$6$ |
$(1,3,6,2,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
