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^{3} + 9 x + 76 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 30\cdot 79 + 76\cdot 79^{2} + 21\cdot 79^{3} + 39\cdot 79^{4} + 42\cdot 79^{5} + 39\cdot 79^{6} + 11\cdot 79^{7} + 32\cdot 79^{8} + 8\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 + 37\cdot 79 + 46\cdot 79^{2} + 7\cdot 79^{3} + 39\cdot 79^{4} + 9\cdot 79^{5} + 67\cdot 79^{6} + 13\cdot 79^{7} + 16\cdot 79^{8} + 8\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 a^{2} + 64 a + 74 + \left(46 a^{2} + 75 a + 36\right)\cdot 79 + \left(19 a^{2} + 33 a + 40\right)\cdot 79^{2} + \left(36 a^{2} + 60 a + 48\right)\cdot 79^{3} + \left(26 a^{2} + 3 a + 32\right)\cdot 79^{4} + \left(10 a^{2} + 72 a + 59\right)\cdot 79^{5} + \left(27 a^{2} + 54 a + 38\right)\cdot 79^{6} + \left(10 a^{2} + 56 a + 40\right)\cdot 79^{7} + \left(68 a^{2} + 34 a + 2\right)\cdot 79^{8} + \left(3 a^{2} + 16 a + 24\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 a^{2} + 76 a + 37 + \left(37 a^{2} + 33 a + 18\right)\cdot 79 + \left(25 a^{2} + 25 a + 48\right)\cdot 79^{2} + \left(71 a^{2} + 19 a + 70\right)\cdot 79^{3} + \left(72 a^{2} + 78 a + 13\right)\cdot 79^{4} + \left(31 a^{2} + 58 a + 10\right)\cdot 79^{5} + \left(28 a^{2} + 9 a + 52\right)\cdot 79^{6} + \left(67 a^{2} + 34 a + 71\right)\cdot 79^{7} + \left(29 a^{2} + 51 a + 48\right)\cdot 79^{8} + \left(61 a^{2} + 31 a + 37\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 71 a^{2} + a + 24 + \left(15 a^{2} + 38 a + 45\right)\cdot 79 + \left(29 a^{2} + 77 a + 70\right)\cdot 79^{2} + \left(a^{2} + 22 a + 45\right)\cdot 79^{3} + \left(76 a^{2} + 72 a + 32\right)\cdot 79^{4} + \left(76 a^{2} + 45 a + 43\right)\cdot 79^{5} + \left(25 a^{2} + 35 a + 37\right)\cdot 79^{6} + \left(58 a^{2} + 59 a + 17\right)\cdot 79^{7} + \left(73 a^{2} + 56 a + 75\right)\cdot 79^{8} + \left(a^{2} + 77 a + 75\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 a^{2} + 2 a + 76 + \left(25 a^{2} + 7 a + 22\right)\cdot 79 + \left(24 a^{2} + 55 a + 41\right)\cdot 79^{2} + \left(6 a^{2} + 36 a + 75\right)\cdot 79^{3} + \left(9 a^{2} + 7 a + 25\right)\cdot 79^{4} + \left(49 a^{2} + 53 a + 34\right)\cdot 79^{5} + \left(24 a^{2} + 33 a + 29\right)\cdot 79^{6} + \left(32 a^{2} + 64 a + 19\right)\cdot 79^{7} + \left(54 a^{2} + 49 a + 38\right)\cdot 79^{8} + \left(15 a^{2} + 48 a\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 60 + 23\cdot 79 + 27\cdot 79^{2} + 48\cdot 79^{3} + 69\cdot 79^{4} + 25\cdot 79^{5} + 67\cdot 79^{6} + 9\cdot 79^{7} + 60\cdot 79^{8} + 24\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 13 a^{2} + 9 a + 23 + \left(57 a^{2} + 47 a + 20\right)\cdot 79 + \left(50 a^{2} + 11 a + 69\right)\cdot 79^{2} + \left(15 a^{2} + 20 a + 3\right)\cdot 79^{3} + \left(8 a^{2} + 26 a + 2\right)\cdot 79^{4} + \left(42 a^{2} + 52 a + 13\right)\cdot 79^{5} + \left(4 a^{2} + 43 a + 61\right)\cdot 79^{6} + \left(63 a^{2} + 50 a + 40\right)\cdot 79^{7} + \left(54 a^{2} + 35 a + 1\right)\cdot 79^{8} + \left(62 a^{2} + 41 a + 61\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 5 a^{2} + 6 a + 54 + \left(54 a^{2} + 35 a + 1\right)\cdot 79 + \left(8 a^{2} + 33 a + 54\right)\cdot 79^{2} + \left(27 a^{2} + 77 a + 72\right)\cdot 79^{3} + \left(44 a^{2} + 48 a + 60\right)\cdot 79^{4} + \left(26 a^{2} + 33 a + 77\right)\cdot 79^{5} + \left(47 a^{2} + 59 a + 1\right)\cdot 79^{6} + \left(5 a^{2} + 50 a + 12\right)\cdot 79^{7} + \left(35 a^{2} + 8 a + 41\right)\cdot 79^{8} + \left(12 a^{2} + 21 a + 75\right)\cdot 79^{9} +O\left(79^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,7,2)(4,6,5)$ |
| $(1,4)(2,6)(5,7)$ |
| $(1,7,2)(3,9,8)(4,5,6)$ |
| $(1,3,6)(2,8,5)(4,7,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $9$ |
$2$ |
$(1,8)(2,9)(3,7)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,7,2)(3,9,8)(4,5,6)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $1$ |
$3$ |
$(1,2,7)(3,8,9)(4,6,5)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $6$ |
$3$ |
$(1,3,6)(2,8,5)(4,7,9)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,3,5)(2,8,4)(6,7,9)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,7,2)(4,6,5)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,4,3)(2,6,8)(5,9,7)$ |
$0$ |
$0$ |
| $9$ |
$6$ |
$(1,3,2,8,7,9)(4,5,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $9$ |
$6$ |
$(1,9,7,8,2,3)(4,6,5)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
