Basic invariants
Dimension: | $2$ |
Group: | $D_{15}$ |
Conductor: | \(2159\)\(\medspace = 17 \cdot 127 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 15.1.218659573334046061397519.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $D_{15}$ |
Parity: | odd |
Determinant: | 1.2159.2t1.a.a |
Projective image: | $D_{15}$ |
Projective stem field: | Galois closure of 15.1.218659573334046061397519.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{15} - 15 x^{12} + 14 x^{11} + 101 x^{10} + 110 x^{9} - 304 x^{8} - 603 x^{7} - 60 x^{6} + 921 x^{5} + 1057 x^{4} + 735 x^{3} + 384 x^{2} + 117 x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \( x^{5} + x + 42 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 9 a^{3} + 37 a^{2} + 34 a + 5 + \left(29 a^{4} + 19 a^{3} + 34 a^{2} + 45 a + 1\right)\cdot 47 + \left(20 a^{4} + 28 a^{3} + 20 a^{2} + 30 a + 12\right)\cdot 47^{2} + \left(46 a^{4} + 42 a^{3} + 8 a^{2} + 43 a + 23\right)\cdot 47^{3} + \left(a^{4} + 29 a^{3} + 4 a^{2} + 15 a + 24\right)\cdot 47^{4} +O(47^{5})\)
$r_{ 2 }$ |
$=$ |
\( 5 a^{4} + 30 a^{3} + 43 a^{2} + 43 a + 33 + \left(43 a^{4} + 17 a^{3} + 11 a^{2} + 30 a + 33\right)\cdot 47 + \left(31 a^{4} + 10 a^{2} + 36 a + 35\right)\cdot 47^{2} + \left(21 a^{4} + 4 a^{3} + 18 a^{2} + 38 a + 20\right)\cdot 47^{3} + \left(41 a^{4} + 38 a^{3} + 34 a^{2} + 8 a + 44\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 6 a^{4} + 24 a^{3} + 26 a^{2} + 27 a + 15 + \left(3 a^{4} + 34 a^{3} + 15 a^{2} + 45 a + 39\right)\cdot 47 + \left(30 a^{4} + 36 a^{3} + 7 a^{2} + 35 a + 24\right)\cdot 47^{2} + \left(24 a^{4} + 31 a^{3} + 25 a^{2} + 18 a + 32\right)\cdot 47^{3} + \left(22 a^{4} + 45 a^{3} + 23 a^{2} + 15 a + 38\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 9 a^{4} + 10 a^{3} + 21 a^{2} + 25 a + 31 + \left(41 a^{4} + 36 a^{3} + 41 a^{2} + 35 a + 29\right)\cdot 47 + \left(8 a^{4} + 44 a^{3} + 40 a^{2} + 24 a + 2\right)\cdot 47^{2} + \left(a^{4} + 12 a^{3} + 14 a^{2} + 22 a + 34\right)\cdot 47^{3} + \left(24 a^{4} + 35 a^{3} + 34 a^{2} + 4 a + 32\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 17 a^{4} + 24 a^{3} + 35 a^{2} + 32 a + 36 + \left(2 a^{4} + 15 a^{3} + 15 a^{2} + 26 a + 24\right)\cdot 47 + \left(3 a^{4} + a^{2} + 3 a + 15\right)\cdot 47^{2} + \left(22 a^{4} + 30 a^{3} + 6 a^{2} + 26 a + 9\right)\cdot 47^{3} + \left(26 a^{4} + 28 a^{3} + 9 a^{2} + 18 a + 15\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 19 a^{4} + 43 a^{3} + 20 a^{2} + 37 a + 39 + \left(28 a^{4} + 17 a^{3} + 13 a^{2} + 24 a + 28\right)\cdot 47 + \left(17 a^{4} + 10 a^{3} + 2 a^{2} + 15 a + 9\right)\cdot 47^{2} + \left(43 a^{4} + 32 a^{3} + 33 a^{2} + 12 a + 30\right)\cdot 47^{3} + \left(10 a^{4} + 40 a^{3} + 4 a^{2} + 12 a + 31\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 21 a^{4} + 42 a^{3} + 27 a^{2} + 11 a + 27 + \left(37 a^{4} + 2 a^{3} + 37 a^{2} + 43 a + 38\right)\cdot 47 + \left(a^{4} + 13 a^{3} + 20 a^{2} + 30 a + 39\right)\cdot 47^{2} + \left(5 a^{4} + 40 a^{3} + 34 a^{2} + 24 a + 16\right)\cdot 47^{3} + \left(40 a^{4} + 2 a^{3} + 20 a^{2} + 13 a + 43\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 23 a^{4} + 19 a^{3} + 13 a^{2} + 38 a + 22 + \left(8 a^{4} + 35 a^{3} + 2 a^{2} + 5 a + 1\right)\cdot 47 + \left(17 a^{4} + 46 a^{3} + 5 a^{2} + 28 a + 8\right)\cdot 47^{2} + \left(22 a^{4} + 21 a^{3} + 6 a^{2} + 5 a\right)\cdot 47^{3} + \left(13 a^{4} + 6 a^{3} + a^{2} + 25 a + 33\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 9 }$ |
$=$ |
\( 24 a^{4} + 19 a^{3} + 44 a^{2} + 22 a + 20 + \left(9 a^{4} + 39 a^{3} + 9 a^{2} + 42 a + 44\right)\cdot 47 + \left(9 a^{4} + 18 a^{3} + 21 a^{2} + 34 a + 26\right)\cdot 47^{2} + \left(25 a^{4} + 29 a^{3} + 32 a^{2} + 44 a + 23\right)\cdot 47^{3} + \left(31 a^{4} + 10 a^{3} + 41 a^{2} + 5 a + 36\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 10 }$ |
$=$ |
\( 25 a^{4} + 40 a^{3} + 16 a^{2} + 19 a + 25 + \left(a^{4} + 13 a^{3} + 19 a^{2} + 36 a + 35\right)\cdot 47 + \left(12 a^{4} + 46 a^{3} + 35 a^{2} + 6 a + 42\right)\cdot 47^{2} + \left(3 a^{4} + 12 a^{3} + 22 a^{2} + 29 a + 16\right)\cdot 47^{3} + \left(26 a^{4} + 27 a^{3} + 3 a^{2} + 19 a + 34\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 11 }$ |
$=$ |
\( 32 a^{4} + 13 a^{3} + 42 a + 1 + \left(2 a^{4} + 23 a^{3} + 37 a^{2} + 12 a + 25\right)\cdot 47 + \left(8 a^{4} + 12 a^{3} + 45 a^{2} + 33 a + 19\right)\cdot 47^{2} + \left(21 a^{4} + 2 a^{3} + 6 a^{2} + 5 a + 27\right)\cdot 47^{3} + \left(13 a^{3} + 36 a^{2} + 27 a + 22\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 12 }$ |
$=$ |
\( 32 a^{4} + 13 a^{3} + 20 a^{2} + 10 a + 1 + \left(15 a^{4} + 37 a^{3} + 24 a^{2} + 5 a + 26\right)\cdot 47 + \left(10 a^{4} + 22 a^{3} + 31 a^{2} + 2\right)\cdot 47^{2} + \left(42 a^{4} + 13 a^{3} + 44 a^{2} + 36 a + 16\right)\cdot 47^{3} + \left(22 a^{4} + 36 a^{3} + 25 a^{2} + 38 a + 12\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 13 }$ |
$=$ |
\( 37 a^{4} + 25 a^{3} + 26 a^{2} + 19 a + 5 + \left(17 a^{4} + 29 a^{3} + 14 a^{2} + 43 a + 37\right)\cdot 47 + \left(8 a^{4} + 11 a^{3} + 10 a^{2} + 28 a + 19\right)\cdot 47^{2} + \left(33 a^{4} + 26 a^{3} + 30 a^{2} + 20 a + 46\right)\cdot 47^{3} + \left(30 a^{4} + 9 a^{3} + 21 a^{2} + 31 a + 27\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 14 }$ |
$=$ |
\( 38 a^{4} + 26 a^{3} + a^{2} + 38 a + 3 + \left(46 a^{3} + 19 a^{2} + 25 a + 28\right)\cdot 47 + \left(21 a^{4} + 24 a^{3} + 34 a^{2} + 2 a + 17\right)\cdot 47^{2} + \left(17 a^{4} + 35 a^{3} + 30 a^{2} + 14 a + 17\right)\cdot 47^{3} + \left(5 a^{4} + 43 a^{3} + 20 a^{2} + 3 a + 34\right)\cdot 47^{4} +O(47^{5})\)
| $r_{ 15 }$ |
$=$ |
\( 41 a^{4} + 39 a^{3} + 26 a + 19 + \left(40 a^{4} + 6 a^{3} + 32 a^{2} + 45 a + 29\right)\cdot 47 + \left(34 a^{4} + 11 a^{3} + 41 a^{2} + 15 a + 4\right)\cdot 47^{2} + \left(46 a^{4} + 40 a^{3} + 14 a^{2} + 33 a + 14\right)\cdot 47^{3} + \left(30 a^{4} + 7 a^{3} + 41 a + 38\right)\cdot 47^{4} +O(47^{5})\)
| |
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 value |
$1$ | $1$ | $()$ | $2$ |
$15$ | $2$ | $(1,10)(2,8)(3,12)(4,15)(5,9)(7,11)(13,14)$ | $0$ |
$2$ | $3$ | $(1,9,8)(2,5,10)(3,11,4)(6,14,13)(7,12,15)$ | $-1$ |
$2$ | $5$ | $(1,4,15,10,6)(2,14,9,3,7)(5,13,8,11,12)$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ |
$2$ | $5$ | $(1,15,6,4,10)(2,9,7,14,3)(5,8,12,13,11)$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ |
$2$ | $15$ | $(1,3,12,10,14,8,4,7,5,6,9,11,15,2,13)$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ |
$2$ | $15$ | $(1,12,14,4,5,9,15,13,3,10,8,7,6,11,2)$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ |
$2$ | $15$ | $(1,14,5,15,3,8,6,2,12,4,9,13,10,7,11)$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ |
$2$ | $15$ | $(1,7,13,4,2,8,15,14,11,10,9,12,6,3,5)$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ |
The blue line marks the conjugacy class containing complex conjugation.