Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$: $ x^{2} + 60 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 18\cdot 61 + 32\cdot 61^{2} + 41\cdot 61^{3} + 51\cdot 61^{4} + 43\cdot 61^{5} + 50\cdot 61^{6} + 57\cdot 61^{7} + 26\cdot 61^{8} + 40\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 51 a + 55 + \left(35 a + 4\right)\cdot 61 + \left(2 a + 52\right)\cdot 61^{2} + \left(56 a + 27\right)\cdot 61^{3} + \left(10 a + 2\right)\cdot 61^{4} + \left(32 a + 47\right)\cdot 61^{5} + \left(50 a + 19\right)\cdot 61^{6} + \left(9 a + 4\right)\cdot 61^{7} + \left(22 a + 3\right)\cdot 61^{8} + \left(12 a + 53\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 a + 45 + \left(25 a + 50\right)\cdot 61 + \left(58 a + 18\right)\cdot 61^{2} + \left(4 a + 20\right)\cdot 61^{3} + \left(50 a + 18\right)\cdot 61^{4} + \left(28 a + 7\right)\cdot 61^{5} + \left(10 a + 38\right)\cdot 61^{6} + \left(51 a + 24\right)\cdot 61^{7} + \left(38 a + 15\right)\cdot 61^{8} + \left(48 a + 43\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 + 42\cdot 61 + 28\cdot 61^{2} + 19\cdot 61^{3} + 9\cdot 61^{4} + 17\cdot 61^{5} + 10\cdot 61^{6} + 3\cdot 61^{7} + 34\cdot 61^{8} + 20\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 10 a + 6 + \left(25 a + 56\right)\cdot 61 + \left(58 a + 8\right)\cdot 61^{2} + \left(4 a + 33\right)\cdot 61^{3} + \left(50 a + 58\right)\cdot 61^{4} + \left(28 a + 13\right)\cdot 61^{5} + \left(10 a + 41\right)\cdot 61^{6} + \left(51 a + 56\right)\cdot 61^{7} + \left(38 a + 57\right)\cdot 61^{8} + \left(48 a + 7\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 a + 16 + \left(35 a + 10\right)\cdot 61 + \left(2 a + 42\right)\cdot 61^{2} + \left(56 a + 40\right)\cdot 61^{3} + \left(10 a + 42\right)\cdot 61^{4} + \left(32 a + 53\right)\cdot 61^{5} + \left(50 a + 22\right)\cdot 61^{6} + \left(9 a + 36\right)\cdot 61^{7} + \left(22 a + 45\right)\cdot 61^{8} + \left(12 a + 17\right)\cdot 61^{9} +O\left(61^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,4)$ |
| $(1,3,2)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-3$ |
| $3$ | $2$ | $(1,4)(3,6)$ | $-1$ |
| $3$ | $2$ | $(1,4)$ | $1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $1$ |
| $8$ | $3$ | $(1,3,2)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,3,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,5,4,2)(3,6)$ | $1$ |
| $8$ | $6$ | $(1,6,5,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
