Basic invariants
| Dimension: | $2$ |
| Group: | $D_{15}$ |
| Conductor: | \(751\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 15.1.134734730815558692751.1 |
| Galois orbit size: | $4$ |
| Smallest permutation container: | $D_{15}$ |
| Parity: | odd |
| Projective image: | $D_{15}$ |
| Projective field: | Galois closure of 15.1.134734730815558692751.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$:
\( x^{5} + 8x + 40 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( a^{4} + 27 a^{3} + 38 a^{2} + 42 a + 1 + \left(21 a^{4} + 5 a^{3} + 3 a^{2} + 27 a + 5\right)\cdot 43 + \left(34 a^{4} + 2 a^{3} + 10 a^{2} + 40 a + 12\right)\cdot 43^{2} + \left(5 a^{4} + 35 a^{3} + 5 a^{2} + 10 a + 10\right)\cdot 43^{3} + \left(18 a^{4} + 21 a^{3} + 37 a^{2} + 30 a + 31\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 a^{4} + 36 a^{3} + 32 a^{2} + 17 a + 18 + \left(7 a^{4} + 30 a^{3} + 39 a^{2} + 21 a + 9\right)\cdot 43 + \left(29 a^{4} + 17 a^{3} + 10 a^{2} + 4 a + 39\right)\cdot 43^{2} + \left(27 a^{4} + 31 a^{3} + 40 a^{2} + 20 a + 18\right)\cdot 43^{3} + \left(19 a^{4} + 38 a^{3} + 35 a^{2} + 17 a + 16\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a^{4} + 7 a^{3} + 15 a^{2} + 36 a + 3 + \left(24 a^{4} + 28 a^{3} + 21 a^{2} + 28 a + 36\right)\cdot 43 + \left(18 a^{4} + 27 a^{3} + 33 a + 1\right)\cdot 43^{2} + \left(15 a^{4} + 24 a^{3} + 30 a^{2} + 3 a\right)\cdot 43^{3} + \left(27 a^{4} + 24 a^{3} + 10 a^{2} + 24 a + 8\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 5 a^{4} + 27 a^{3} + 7 a^{2} + 9 a + 33 + \left(8 a^{4} + 27 a^{3} + 20 a^{2} + 41 a + 41\right)\cdot 43 + \left(4 a^{4} + 30 a^{3} + 2 a^{2} + 15 a + 16\right)\cdot 43^{2} + \left(27 a^{4} + 42 a^{3} + 9 a^{2} + 13 a + 32\right)\cdot 43^{3} + \left(19 a^{4} + 41 a^{3} + 16 a^{2} + 4 a + 7\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 6 a^{4} + 40 a^{3} + 30 a^{2} + 31 a + 18 + \left(18 a^{4} + 15 a^{3} + 5 a^{2} + 2 a + 6\right)\cdot 43 + \left(34 a^{4} + 41 a^{3} + 21 a^{2} + 14 a\right)\cdot 43^{2} + \left(42 a^{4} + 37 a^{3} + 14 a^{2} + 13 a + 29\right)\cdot 43^{3} + \left(9 a^{4} + 22 a^{3} + 4 a^{2} + 10 a + 8\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 11 a^{4} + 9 a^{3} + a^{2} + a + 22 + \left(38 a^{4} + 18 a^{3} + 30 a^{2} + 8 a + 3\right)\cdot 43 + \left(11 a^{4} + 24 a^{3} + 21 a^{2} + 13 a + 5\right)\cdot 43^{2} + \left(3 a^{3} + 12 a^{2} + 29 a + 35\right)\cdot 43^{3} + \left(42 a^{4} + 35 a^{3} + 37 a^{2} + 42 a + 37\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 11 a^{4} + 26 a^{3} + 40 a^{2} + 29 a + 37 + \left(21 a^{4} + 32 a^{3} + 33 a^{2} + 20 a + 39\right)\cdot 43 + \left(38 a^{4} + 6 a^{3} + 30 a^{2} + 25 a + 12\right)\cdot 43^{2} + \left(14 a^{4} + 12 a^{3} + 21 a^{2} + 31 a + 40\right)\cdot 43^{3} + \left(6 a^{3} + 28 a^{2} + 10 a + 21\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 12 a^{4} + 7 a^{3} + 37 a^{2} + 6 a + 37 + \left(14 a^{4} + 11 a^{3} + 39 a^{2} + 35 a + 4\right)\cdot 43 + \left(19 a^{4} + 39 a^{3} + 25 a^{2} + 9 a + 1\right)\cdot 43^{2} + \left(3 a^{4} + 31 a^{3} + 26 a^{2} + 34 a + 21\right)\cdot 43^{3} + \left(27 a^{4} + 5 a^{3} + 13 a^{2} + 14 a + 2\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 9 }$ | $=$ |
\( 17 a^{4} + 6 a^{3} + 16 a^{2} + 27 a + 11 + \left(21 a^{4} + 36 a^{3} + 4 a^{2} + 30 a + 27\right)\cdot 43 + \left(25 a^{4} + 39 a^{3} + 13 a^{2} + 32 a + 37\right)\cdot 43^{2} + \left(36 a^{4} + 4 a^{3} + 6 a^{2} + 36 a + 40\right)\cdot 43^{3} + \left(27 a^{4} + 25 a^{3} + 14 a^{2} + 38 a + 36\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 10 }$ | $=$ |
\( 26 a^{4} + 37 a^{3} + 12 a^{2} + 11 a + 4 + \left(29 a^{4} + 8 a^{3} + 7 a^{2} + 32 a + 33\right)\cdot 43 + \left(39 a^{4} + 20 a^{3} + 15 a + 37\right)\cdot 43^{2} + \left(42 a^{4} + a^{3} + 16 a^{2} + 21\right)\cdot 43^{3} + \left(33 a^{4} + 28 a^{3} + a^{2} + 33 a + 39\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 11 }$ | $=$ |
\( 27 a^{4} + 17 a^{2} + 20 a + 32 + \left(21 a^{4} + a^{3} + 6 a^{2} + 29 a + 28\right)\cdot 43 + \left(37 a^{4} + 29 a^{3} + 6 a^{2} + 28 a + 2\right)\cdot 43^{2} + \left(11 a^{4} + 22 a^{3} + 19 a^{2} + 31 a + 3\right)\cdot 43^{3} + \left(39 a^{4} + 41 a^{3} + 36 a^{2} + 10 a + 24\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 12 }$ | $=$ |
\( 29 a^{4} + 34 a^{3} + 32 a^{2} + 39 a + 34 + \left(a^{4} + 20 a^{3} + 10 a^{2} + 41 a + 18\right)\cdot 43 + \left(35 a^{3} + 31 a^{2} + 28 a + 32\right)\cdot 43^{2} + \left(33 a^{4} + 39 a^{3} + 37 a^{2} + 28 a + 3\right)\cdot 43^{3} + \left(2 a^{4} + 3 a^{3} + 24 a^{2} + 26 a + 36\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 13 }$ | $=$ |
\( 31 a^{4} + 33 a^{3} + 8 a^{2} + 15 a + 6 + \left(4 a^{3} + 5 a^{2} + 37 a + 41\right)\cdot 43 + \left(13 a^{4} + 34 a^{3} + 2 a^{2} + 19 a + 17\right)\cdot 43^{2} + \left(22 a^{4} + 38 a^{3} + 16 a^{2} + 35\right)\cdot 43^{3} + \left(24 a^{4} + 14 a^{3} + 20 a^{2} + 2 a + 32\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 14 }$ | $=$ |
\( 33 a^{4} + 9 a^{3} + 21 a^{2} + 41 a + 8 + \left(10 a^{4} + 30 a^{3} + a^{2} + 15 a + 8\right)\cdot 43 + \left(20 a^{4} + 27 a^{3} + 40 a^{2} + 36 a + 24\right)\cdot 43^{2} + \left(18 a^{3} + 3 a^{2} + 25 a + 10\right)\cdot 43^{3} + \left(39 a^{4} + 19 a^{3} + 16 a^{2} + 14 a + 27\right)\cdot 43^{4} +O(43^{5})\)
|
| $r_{ 15 }$ | $=$ |
\( 40 a^{4} + 3 a^{3} + 38 a^{2} + 20 a + 42 + \left(19 a^{4} + 29 a^{3} + 27 a^{2} + 13 a + 39\right)\cdot 43 + \left(17 a^{4} + 10 a^{3} + 41 a^{2} + 24 a + 15\right)\cdot 43^{2} + \left(16 a^{4} + 41 a^{3} + 41 a^{2} + 20 a + 41\right)\cdot 43^{3} + \left(12 a^{4} + 13 a^{3} + 3 a^{2} + 20 a + 12\right)\cdot 43^{4} +O(43^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 15 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 15 }$ | Character values | |||
| $c1$ | $c2$ | $c3$ | $c4$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ | $2$ | $2$ |
| $15$ | $2$ | $(1,10)(2,12)(4,14)(5,13)(6,7)(8,15)(9,11)$ | $0$ | $0$ | $0$ | $0$ |
| $2$ | $3$ | $(1,13,7)(2,8,11)(3,4,14)(5,10,6)(9,15,12)$ | $-1$ | $-1$ | $-1$ | $-1$ |
| $2$ | $5$ | $(1,14,6,12,8)(2,7,4,10,15)(3,5,9,11,13)$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ |
| $2$ | $5$ | $(1,6,8,14,12)(2,4,15,7,10)(3,9,13,5,11)$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ |
| $2$ | $15$ | $(1,10,11,14,15,13,6,2,3,12,7,5,8,4,9)$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ |
| $2$ | $15$ | $(1,11,15,6,3,7,8,9,10,14,13,2,12,5,4)$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ |
| $2$ | $15$ | $(1,15,3,8,10,13,12,4,11,6,7,9,14,2,5)$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ |
| $2$ | $15$ | $(1,2,9,6,4,13,8,15,5,14,7,11,12,10,3)$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ |