Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 101 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 101 }$: $ x^{2} + 97 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 75 + 39\cdot 101 + 89\cdot 101^{2} + 93\cdot 101^{3} + 45\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 65\cdot 101 + 60\cdot 101^{2} + 76\cdot 101^{3} + 21\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 a + 20 + \left(76 a + 42\right)\cdot 101 + \left(27 a + 24\right)\cdot 101^{2} + \left(24 a + 93\right)\cdot 101^{3} + \left(67 a + 9\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 61 a + 79 + \left(24 a + 4\right)\cdot 101 + \left(73 a + 59\right)\cdot 101^{2} + \left(76 a + 61\right)\cdot 101^{3} + \left(33 a + 52\right)\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 3\cdot 101 + 99\cdot 101^{2} + 65\cdot 101^{3} + 87\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 46\cdot 101 + 71\cdot 101^{2} + 12\cdot 101^{3} + 85\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $15$ | $2$ | $(1,2)$ | $1$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
