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:
| $r_{ 1 }$ |
$=$ |
$ 55 a + 5 + \left(38 a + 32\right)\cdot 61 + \left(22 a + 40\right)\cdot 61^{2} + \left(58 a + 48\right)\cdot 61^{3} + \left(54 a + 42\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 29\cdot 61 + 10\cdot 61^{2} + 45\cdot 61^{3} + 3\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 a + 29 + \left(18 a + 60\right)\cdot 61 + \left(50 a + 37\right)\cdot 61^{2} + \left(56 a + 29\right)\cdot 61^{3} + \left(57 a + 17\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 60 + \left(22 a + 15\right)\cdot 61 + \left(38 a + 24\right)\cdot 61^{2} + \left(2 a + 23\right)\cdot 61^{3} + \left(6 a + 39\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 2 + \left(42 a + 45\right)\cdot 61 + \left(10 a + 8\right)\cdot 61^{2} + \left(4 a + 36\right)\cdot 61^{3} + \left(3 a + 18\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $-1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
