Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 60 a + 59 + \left(59 a + 19\right)\cdot 61 + \left(24 a + 26\right)\cdot 61^{2} + \left(58 a + 27\right)\cdot 61^{3} + \left(6 a + 58\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 58 + \left(a + 19\right)\cdot 61 + \left(36 a + 52\right)\cdot 61^{2} + \left(2 a + 60\right)\cdot 61^{3} + \left(54 a + 6\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 20\cdot 61 + 55\cdot 61^{2} + 46\cdot 61^{3} + 13\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 a + 25 + \left(25 a + 3\right)\cdot 61 + \left(33 a + 33\right)\cdot 61^{2} + \left(12 a + 44\right)\cdot 61^{3} + \left(3 a + 41\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 + 20\cdot 61 + 36\cdot 61^{2} + 40\cdot 61^{3} + 29\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 16 + \left(35 a + 38\right)\cdot 61 + \left(27 a + 40\right)\cdot 61^{2} + \left(48 a + 23\right)\cdot 61^{3} + \left(57 a + 32\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,2,6,3,4)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)(3,6)(4,5)$ | $-2$ |
| $15$ | $2$ | $(2,6)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $30$ | $4$ | $(2,4,6,3)$ | $0$ |
| $24$ | $5$ | $(1,6,4,2,5)$ | $-1$ |
| $20$ | $6$ | $(1,5,2,6,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
