Basic invariants
| Dimension: | $1$ |
| Group: | $C_{12}$ |
| Conductor: | \(91\)\(\medspace = 7 \cdot 13 \) |
| Artin number field: | Galois closure of 12.0.61132828589969773.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $C_{12}$ |
| 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 $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{4} + 23x + 6 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 38 a^{3} + 24 a^{2} + \left(19 a^{3} + 8 a^{2} + 22 a + 32\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 38\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 27\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 25\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 38 a^{3} + 24 a^{2} + 34 + \left(19 a^{3} + 8 a^{2} + 22 a + 29\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 12\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 19\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 16\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 38 a^{3} + 24 a^{2} + 16 + \left(19 a^{3} + 8 a^{2} + 22 a + 26\right)\cdot 41 + \left(32 a^{3} + 19 a^{2} + 2 a + 30\right)\cdot 41^{2} + \left(20 a^{3} + 16 a^{2} + 5 a + 3\right)\cdot 41^{3} + \left(26 a^{3} + 37 a^{2} + 26 a + 38\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a^{3} + 34 a^{2} + 20 a + 3 + \left(10 a^{3} + 15 a^{2} + 37 a + 19\right)\cdot 41 + \left(a^{2} + 37 a + 20\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 38\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 16 a^{3} + 34 a^{2} + 20 a + 10 + \left(10 a^{3} + 15 a^{2} + 37 a + 21\right)\cdot 41 + \left(a^{2} + 37 a + 5\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 6\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a + 10\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 16 a^{3} + 34 a^{2} + 20 a + 26 + \left(10 a^{3} + 15 a^{2} + 37 a + 15\right)\cdot 41 + \left(a^{2} + 37 a + 38\right)\cdot 41^{2} + \left(32 a^{3} + 8 a^{2} + 9 a + 22\right)\cdot 41^{3} + \left(7 a^{3} + 20 a^{2} + 36 a + 22\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 24 a^{3} + 36 a^{2} + 30 a + 25 + \left(13 a^{3} + 12 a^{2} + a + 4\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a + 9\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 32\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 34\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 24 a^{3} + 36 a^{2} + 30 a + \left(13 a^{3} + 12 a^{2} + a + 40\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 8\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 6\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 24 a^{3} + 36 a^{2} + 30 a + 18 + \left(13 a^{3} + 12 a^{2} + a + 2\right)\cdot 41 + \left(34 a^{3} + 2 a^{2} + 13 a + 24\right)\cdot 41^{2} + \left(40 a^{2} + 31 a + 23\right)\cdot 41^{3} + \left(8 a^{3} + 36 a^{2} + 8 a + 25\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 10 }$ | $=$ |
\( 4 a^{3} + 29 a^{2} + 32 a + 1 + \left(38 a^{3} + 3 a^{2} + 20 a + 5\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 7\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 37\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 38\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 11 }$ | $=$ |
\( 4 a^{3} + 29 a^{2} + 32 a + 8 + \left(38 a^{3} + 3 a^{2} + 20 a + 7\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 33\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 4\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 7\right)\cdot 41^{4} +O(41^{5})\)
|
| $r_{ 12 }$ | $=$ |
\( 4 a^{3} + 29 a^{2} + 32 a + 24 + \left(38 a^{3} + 3 a^{2} + 20 a + 1\right)\cdot 41 + \left(14 a^{3} + 18 a^{2} + 28 a + 25\right)\cdot 41^{2} + \left(28 a^{3} + 17 a^{2} + 35 a + 21\right)\cdot 41^{3} + \left(39 a^{3} + 28 a^{2} + 10 a + 19\right)\cdot 41^{4} +O(41^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 12 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 12 }$ | Character values | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $1$ | $1$ | $1$ | $1$ |
| $1$ | $2$ | $(1,7)(2,9)(3,8)(4,10)(5,11)(6,12)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $1$ | $3$ | $(1,3,2)(4,5,6)(7,8,9)(10,11,12)$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ |
| $1$ | $3$ | $(1,2,3)(4,6,5)(7,9,8)(10,12,11)$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ |
| $1$ | $4$ | $(1,11,7,5)(2,10,9,4)(3,12,8,6)$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ |
| $1$ | $4$ | $(1,5,7,11)(2,4,9,10)(3,6,8,12)$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ | $\zeta_{12}^{3}$ | $-\zeta_{12}^{3}$ |
| $1$ | $6$ | $(1,9,3,7,2,8)(4,12,5,10,6,11)$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ |
| $1$ | $6$ | $(1,8,2,7,3,9)(4,11,6,10,5,12)$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ |
| $1$ | $12$ | $(1,4,8,11,2,6,7,10,3,5,9,12)$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}$ |
| $1$ | $12$ | $(1,6,9,11,3,4,7,12,2,5,8,10)$ | $-\zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ |
| $1$ | $12$ | $(1,10,8,5,2,12,7,4,3,11,9,6)$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}$ |
| $1$ | $12$ | $(1,12,9,5,3,10,7,6,2,11,8,4)$ | $\zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ |