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^{3} + 9 x + 76 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 57 a^{2} + 4 a + 45 + \left(69 a^{2} + 37 a + 53\right)\cdot 79 + \left(63 a^{2} + 72 a + 17\right)\cdot 79^{2} + \left(15 a^{2} + 43 a + 8\right)\cdot 79^{3} + \left(28 a^{2} + 61 a + 9\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 76 a^{2} + 39 a + 22 + \left(38 a^{2} + 10 a + 67\right)\cdot 79 + \left(5 a^{2} + 11 a + 29\right)\cdot 79^{2} + \left(70 a^{2} + 39 a + 71\right)\cdot 79^{3} + \left(50 a^{2} + 38\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 a^{2} + 32 a + 62 + \left(60 a^{2} + 12 a + 22\right)\cdot 79 + \left(49 a^{2} + 45 a + 35\right)\cdot 79^{2} + \left(53 a^{2} + 69 a + 46\right)\cdot 79^{3} + \left(39 a^{2} + 57 a + 32\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 68 a^{2} + 27 a + 53 + \left(12 a^{2} + 43 a + 68\right)\cdot 79 + \left(44 a^{2} + 31 a + 24\right)\cdot 79^{2} + \left(71 a^{2} + 39 a + 1\right)\cdot 79^{3} + \left(15 a^{2} + 34 a + 66\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ a^{2} + 67 a + 25 + \left(62 a^{2} + 70 a + 7\right)\cdot 79 + \left(59 a^{2} + 42 a + 72\right)\cdot 79^{2} + \left(18 a^{2} + 18 a + 25\right)\cdot 79^{3} + \left(46 a^{2} + 55 a + 38\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 a^{2} + 12 a + 43 + \left(33 a^{2} + 31 a + 19\right)\cdot 79 + \left(61 a^{2} + 3 a + 26\right)\cdot 79^{2} + \left(33 a^{2} + 75 a + 6\right)\cdot 79^{3} + \left(73 a^{2} + 30 a + 77\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 21 a^{2} + 8 a + 66 + \left(26 a^{2} + 50 a + 29\right)\cdot 79 + \left(34 a^{2} + 42 a + 77\right)\cdot 79^{2} + \left(44 a^{2} + 16 a + 21\right)\cdot 79^{3} + \left(4 a^{2} + 41 a + 25\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 14 a^{2} + 13 a + 45 + \left(27 a^{2} + 25 a + 75\right)\cdot 79 + \left(29 a^{2} + 36 a + 14\right)\cdot 79^{2} + \left(16 a^{2} + 65\right)\cdot 79^{3} + \left(12 a^{2} + 44 a + 43\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 55 a^{2} + 35 a + 34 + \left(64 a^{2} + 35 a + 50\right)\cdot 79 + \left(46 a^{2} + 30 a + 17\right)\cdot 79^{2} + \left(70 a^{2} + 13 a + 69\right)\cdot 79^{3} + \left(44 a^{2} + 69 a + 63\right)\cdot 79^{4} +O\left(79^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,7,5)(2,4,8)(3,9,6)$ |
| $(1,5)(2,3)(4,6)(8,9)$ |
| $(1,6,2,7,3,4,5,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
$2$ |
| $9$ |
$2$ |
$(1,5)(2,3)(4,6)(8,9)$ |
$0$ |
$0$ |
$0$ |
| $2$ |
$3$ |
$(1,7,5)(2,4,8)(3,9,6)$ |
$-1$ |
$-1$ |
$-1$ |
| $2$ |
$9$ |
$(1,6,2,7,3,4,5,9,8)$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
| $2$ |
$9$ |
$(1,2,3,5,8,6,7,4,9)$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
| $2$ |
$9$ |
$(1,3,8,7,9,2,5,6,4)$ |
$-\zeta_{9}^{5} - \zeta_{9}^{2} + \zeta_{9}$ |
$-\zeta_{9}^{4} + \zeta_{9}^{2} - \zeta_{9}$ |
$\zeta_{9}^{5} + \zeta_{9}^{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
