Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$: $ x^{2} + 78 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 38 + \left(41 a + 19\right)\cdot 79 + \left(74 a + 66\right)\cdot 79^{2} + \left(50 a + 19\right)\cdot 79^{3} + \left(15 a + 70\right)\cdot 79^{4} + \left(76 a + 23\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 72 a + 12 + \left(26 a + 75\right)\cdot 79 + \left(20 a + 7\right)\cdot 79^{2} + \left(66 a + 45\right)\cdot 79^{3} + \left(69 a + 4\right)\cdot 79^{4} + \left(16 a + 13\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 68 + 12\cdot 79 + 16\cdot 79^{2} + 44\cdot 79^{3} + 43\cdot 79^{4} + 6\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 73 a + 44 + \left(37 a + 54\right)\cdot 79 + \left(4 a + 20\right)\cdot 79^{2} + \left(28 a + 75\right)\cdot 79^{3} + \left(63 a + 34\right)\cdot 79^{4} + \left(2 a + 5\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 70 + 44\cdot 79 + 45\cdot 79^{2} + 40\cdot 79^{3} + 75\cdot 79^{4} + 69\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 a + 5 + \left(52 a + 30\right)\cdot 79 + \left(58 a + 1\right)\cdot 79^{2} + \left(12 a + 12\right)\cdot 79^{3} + \left(9 a + 8\right)\cdot 79^{4} + \left(62 a + 39\right)\cdot 79^{5} +O\left(79^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,5,6)$ |
| $(4,6)$ |
| $(1,4)(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)(4,6)$ | $-1$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,6)$ | $1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $-1$ |
| $8$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(1,6,2,4)$ | $1$ |
| $6$ | $4$ | $(1,6,2,4)(3,5)$ | $-1$ |
| $8$ | $6$ | $(1,6,5,2,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
