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:
| $r_{ 1 }$ |
$=$ |
$ 37 a + 55 + \left(3 a + 40\right)\cdot 61 + \left(17 a + 25\right)\cdot 61^{2} + \left(20 a + 53\right)\cdot 61^{3} + \left(10 a + 27\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 11\cdot 61 + 18\cdot 61^{2} + 21\cdot 61^{3} + 48\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 49\cdot 61 + 16\cdot 61^{2} + 20\cdot 61^{3} + 39\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 30 + \left(3 a + 36\right)\cdot 61 + \left(27 a + 29\right)\cdot 61^{2} + 57 a\cdot 61^{3} + \left(13 a + 16\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 a + 31 + \left(57 a + 7\right)\cdot 61 + \left(43 a + 39\right)\cdot 61^{2} + \left(40 a + 56\right)\cdot 61^{3} + \left(50 a + 17\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 59 a + 32 + \left(57 a + 37\right)\cdot 61 + \left(33 a + 53\right)\cdot 61^{2} + \left(3 a + 30\right)\cdot 61^{3} + \left(47 a + 33\right)\cdot 61^{4} +O\left(61^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)(2,3)(4,5)$ |
| $(1,3,2,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,6)(2,3)(4,5)$ | $1$ |
| $15$ | $2$ | $(2,3)(4,6)$ | $1$ |
| $20$ | $3$ | $(1,2,6)(3,4,5)$ | $-1$ |
| $30$ | $4$ | $(2,4,3,6)$ | $-1$ |
| $24$ | $5$ | $(1,4,6,5,2)$ | $0$ |
| $20$ | $6$ | $(1,3,2,4,6,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
