Basic invariants
| Dimension: | $1$ |
| Group: | $C_{10}$ |
| Conductor: | \(75\)\(\medspace = 3 \cdot 5^{2} \) |
| Artin number field: | Galois closure of 10.0.37078857421875.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_{10}$ |
| Parity: | odd |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{5} + 4x + 11 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 3 a^{4} + 10 a^{3} + 12 a^{2} + 6 a + 7 + \left(12 a^{4} + 6 a^{2} + 11 a + 10\right)\cdot 13 + \left(6 a^{4} + 5 a^{2} + 4 a + 11\right)\cdot 13^{2} + \left(2 a^{4} + 7 a^{2} + 10 a + 2\right)\cdot 13^{3} + \left(8 a^{4} + 3 a^{3} + 5 a^{2} + 6 a\right)\cdot 13^{4} + \left(2 a^{4} + 3 a^{3} + 11 a^{2} + 11 a + 11\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 a^{4} + 2 a^{3} + 4 a^{2} + 12 a + 10 + \left(5 a^{4} + a^{3} + 2 a^{2} + 3 a + 12\right)\cdot 13 + \left(6 a^{4} + 3 a^{2} + 11 a + 4\right)\cdot 13^{2} + \left(9 a^{4} + a^{3} + 7 a + 4\right)\cdot 13^{3} + \left(7 a^{4} + 4 a^{3} + 7 a^{2} + 9 a + 1\right)\cdot 13^{4} + \left(2 a^{4} + 11 a^{3} + 3\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a^{4} + 8 a^{2} + 2 + \left(a^{4} + 6 a^{3} + 9 a^{2} + 3 a + 6\right)\cdot 13 + \left(10 a^{4} + 5 a^{3} + 7 a^{2} + 6\right)\cdot 13^{2} + \left(7 a^{4} + 3 a^{3} + 5 a^{2} + 12 a + 1\right)\cdot 13^{3} + \left(7 a^{4} + 6 a^{3} + 3 a^{2} + 12 a + 6\right)\cdot 13^{4} + \left(7 a^{4} + 8 a^{3} + 11 a^{2} + 9 a + 3\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a^{4} + 7 a^{3} + 3 a^{2} + 3 + \left(7 a^{4} + 5 a^{3} + 12 a^{2} + 10 a\right)\cdot 13 + \left(9 a^{4} + 2 a^{3} + 11 a^{2} + 12 a + 2\right)\cdot 13^{2} + \left(5 a^{2} + a + 10\right)\cdot 13^{3} + \left(4 a^{4} + 11 a^{3} + 2 a^{2} + 11 a + 7\right)\cdot 13^{4} + \left(11 a^{4} + 12 a^{3} + 6 a + 7\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 9 a^{4} + 4 a^{3} + 10 a^{2} + 5 a + 8 + \left(4 a^{4} + 8 a^{3} + 9 a^{2} + 9 a + 4\right)\cdot 13 + \left(4 a^{4} + 3 a^{3} + 8 a^{2} + 7 a + 11\right)\cdot 13^{2} + \left(2 a^{4} + 4 a^{3} + 11 a^{2} + 12 a + 4\right)\cdot 13^{3} + \left(4 a^{4} + 8 a^{3} + 4 a^{2} + 9 a\right)\cdot 13^{4} + \left(7 a^{4} + 3 a^{3} + 2 a^{2} + 8 a\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a^{4} + 7 a^{3} + 8 a + 1 + \left(10 a^{4} + 3 a^{3} + a^{2} + 12 a + 10\right)\cdot 13 + \left(6 a^{4} + 7 a^{3} + 8 a^{2} + 4 a + 8\right)\cdot 13^{2} + \left(2 a^{4} + 4 a^{3} + 2 a^{2} + 9 a + 10\right)\cdot 13^{3} + \left(11 a^{4} + a^{3} + 2 a^{2} + 2 a + 9\right)\cdot 13^{4} + \left(12 a^{4} + 5 a^{3} + 10 a^{2} + 11 a + 4\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 11 a^{4} + 6 a^{3} + 12 a^{2} + 10 a + 4 + \left(12 a^{3} + 11 a^{2} + 5\right)\cdot 13 + \left(12 a^{4} + 11 a^{3} + 6 a^{2} + 6 a + 7\right)\cdot 13^{2} + \left(9 a^{4} + 2 a^{3} + 7 a^{2} + 6 a\right)\cdot 13^{3} + \left(12 a^{4} + 9 a^{3} + 3 a + 7\right)\cdot 13^{4} + \left(2 a^{4} + 6 a^{3} + 3 a^{2} + 9 a + 9\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 2 a^{4} + 8 a^{3} + 9 a^{2} + 9 + \left(12 a^{4} + 2 a^{3} + 4 a^{2} + 4 a + 7\right)\cdot 13 + \left(9 a^{4} + 6 a^{3} + 6 a^{2} + 5 a\right)\cdot 13^{2} + \left(a^{4} + 6 a^{3} + 2 a^{2} + 11 a + 3\right)\cdot 13^{3} + \left(10 a^{4} + 5 a^{3} + 2 a^{2} + 1\right)\cdot 13^{4} + \left(9 a^{4} + a^{3} + 2 a^{2} + 2 a\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 9 }$ | $=$ |
\( 5 a^{4} + 8 a^{3} + 11 a + 3 + \left(6 a^{4} + 9 a^{3} + 3 a^{2} + 8 a + 2\right)\cdot 13 + \left(2 a^{4} + 11 a^{3} + 9 a^{2} + 6 a\right)\cdot 13^{2} + \left(4 a^{4} + 8 a^{3} + 11 a^{2} + 9 a + 3\right)\cdot 13^{3} + \left(4 a^{4} + 9 a^{3} + a^{2} + 11 a + 11\right)\cdot 13^{4} + \left(11 a^{4} + 5 a^{3} + 7 a^{2} + 8 a + 12\right)\cdot 13^{5} +O(13^{6})\)
|
| $r_{ 10 }$ | $=$ |
\( 4 a^{4} + 7 a^{2} + 5 + \left(3 a^{4} + 2 a^{3} + 3 a^{2} + a + 5\right)\cdot 13 + \left(9 a^{4} + 3 a^{3} + 10 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(10 a^{4} + 7 a^{3} + 9 a^{2} + 9 a + 10\right)\cdot 13^{3} + \left(7 a^{4} + 6 a^{3} + 8 a^{2} + 8 a + 6\right)\cdot 13^{4} + \left(9 a^{4} + 6 a^{3} + 3 a^{2} + 8 a + 12\right)\cdot 13^{5} +O(13^{6})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 10 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 10 }$ | Character values | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $2$ | $(1,5)(2,7)(3,10)(4,8)(6,9)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $5$ | $(1,10,6,4,2)(3,9,8,7,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ |
| $1$ | $5$ | $(1,6,2,10,4)(3,8,5,9,7)$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,4,10,2,6)(3,7,9,5,8)$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,2,4,6,10)(3,5,7,8,9)$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ |
| $1$ | $10$ | $(1,8,10,7,6,5,4,3,2,9)$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
| $1$ | $10$ | $(1,7,4,9,10,5,2,8,6,3)$ | $-\zeta_{5}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}^{2}$ |
| $1$ | $10$ | $(1,3,6,8,2,5,10,9,4,7)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}^{3}$ |
| $1$ | $10$ | $(1,9,2,3,4,5,6,7,10,8)$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}$ |