Basic invariants
Dimension: | $2$ |
Group: | $D_{15}$ |
Conductor: | \(439\) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 15.1.3142328914862177479.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $D_{15}$ |
Parity: | odd |
Determinant: | 1.439.2t1.a.a |
Projective image: | $D_{15}$ |
Projective stem field: | Galois closure of 15.1.3142328914862177479.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{15} - 5 x^{14} + 11 x^{13} - 9 x^{12} - 7 x^{11} + 17 x^{10} - x^{9} - 29 x^{8} + 38 x^{7} - 13 x^{6} + \cdots - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{5} + 5x + 17 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a^{3} + 18 a^{2} + 12 a + 9 + \left(11 a^{4} + 4 a^{3} + 3 a^{2} + 6 a + 7\right)\cdot 19 + \left(10 a^{4} + 16 a^{3} + 7 a^{2} + 13 a + 5\right)\cdot 19^{2} + \left(10 a^{4} + 15 a^{3} + 5 a^{2} + 13 a + 14\right)\cdot 19^{3} + \left(17 a^{4} + 3 a^{3} + 6 a^{2} + 4 a + 14\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 2 }$ | $=$ | \( a^{4} + 4 a^{3} + 12 a^{2} + 7 a + 10 + \left(2 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 19 + \left(14 a^{4} + 14 a^{3} + a^{2} + 15 a + 17\right)\cdot 19^{2} + \left(2 a^{4} + 8 a^{3} + 17 a^{2} + 14 a + 17\right)\cdot 19^{3} + \left(14 a^{4} + 10 a^{3} + 12 a^{2} + 4 a + 2\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 3 }$ | $=$ | \( 3 a^{4} + 15 a^{3} + 14 a^{2} + 9 a + 2 + \left(13 a^{4} + 7 a^{3} + 12 a + 16\right)\cdot 19 + \left(18 a^{4} + 5 a^{2} + 11 a + 18\right)\cdot 19^{2} + \left(13 a^{4} + 2 a^{3} + 6 a^{2} + 5 a + 8\right)\cdot 19^{3} + \left(14 a^{4} + 13 a^{3} + 15 a^{2} + 5 a + 3\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 4 }$ | $=$ | \( 4 a^{4} + 10 a^{3} + 15 a^{2} + 16 a + 3 + \left(8 a^{4} + 6 a^{3} + 10 a^{2} + 1\right)\cdot 19 + \left(4 a^{4} + 8 a^{3} + 10 a^{2} + 9 a + 17\right)\cdot 19^{2} + \left(3 a^{4} + 9 a^{3} + 18 a^{2} + 7 a\right)\cdot 19^{3} + \left(12 a^{4} + 7 a^{3} + 9 a^{2} + 6 a + 14\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 5 }$ | $=$ | \( 5 a^{4} + 18 a^{3} + 12 a^{2} + 5 a + 6 + \left(a^{4} + a^{3} + 18 a + 16\right)\cdot 19 + \left(12 a^{4} + 4 a^{3} + 7 a^{2} + 13 a + 9\right)\cdot 19^{2} + \left(4 a^{4} + 15 a^{3} + 11 a^{2} + 11 a + 1\right)\cdot 19^{3} + \left(7 a^{4} + 2 a^{3} + 4 a^{2} + 10 a + 5\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 6 }$ | $=$ | \( 7 a^{4} + 16 a^{2} + 9 a + 15 + \left(9 a^{4} + 15 a^{3} + 14 a^{2} + 15 a + 5\right)\cdot 19 + \left(11 a^{4} + 14 a^{3} + 7 a^{2} + 5 a + 7\right)\cdot 19^{2} + \left(10 a^{4} + 12 a^{3} + 7 a^{2} + 9 a + 11\right)\cdot 19^{3} + \left(14 a^{4} + 3 a^{3} + 17 a^{2} + 4\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 7 }$ | $=$ | \( 8 a^{4} + a^{3} + 8 a^{2} + 3 + \left(15 a^{4} + 18 a^{3} + 11 a^{2} + 16 a + 6\right)\cdot 19 + \left(17 a^{4} + 16 a^{3} + 5 a^{2} + 12 a + 15\right)\cdot 19^{2} + \left(6 a^{4} + 11 a^{3} + 15 a^{2} + 17 a + 18\right)\cdot 19^{3} + \left(18 a^{4} + 8 a^{3} + 7 a^{2} + 15 a + 17\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 8 }$ | $=$ | \( 8 a^{4} + 8 a^{3} + 4 a^{2} + 9 a + \left(13 a^{4} + 4 a^{3} + a^{2} + 16 a + 3\right)\cdot 19 + \left(10 a^{4} + 11 a^{3} + 10 a^{2} + 7 a + 4\right)\cdot 19^{2} + \left(9 a^{4} + 8 a^{3} + 13 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(5 a^{4} + 2 a^{3} + 6 a^{2} + 2 a + 6\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 9 }$ | $=$ | \( 10 a^{4} + a^{3} + 18 a^{2} + 9 a + 7 + \left(15 a^{4} + 12 a^{3} + 11 a^{2} + 5 a + 16\right)\cdot 19 + \left(18 a^{4} + 17 a + 17\right)\cdot 19^{2} + \left(5 a^{4} + 15 a^{3} + 3 a^{2} + 14 a + 6\right)\cdot 19^{3} + \left(17 a^{4} + 3 a^{3} + 12 a + 7\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 10 }$ | $=$ | \( 12 a^{4} + 17 a^{3} + 16 a^{2} + \left(12 a^{4} + 14 a^{3} + 2 a^{2} + 8 a + 14\right)\cdot 19 + \left(9 a^{4} + 3 a^{3} + 4 a^{2} + 1\right)\cdot 19^{2} + \left(12 a^{4} + 8 a^{3} + 8 a^{2} + 13 a + 3\right)\cdot 19^{3} + \left(16 a^{4} + 4 a^{3} + 6 a^{2} + 16 a + 11\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 11 }$ | $=$ | \( 13 a^{4} + 7 a^{3} + 18 a + \left(17 a^{4} + 3 a^{3} + 14 a^{2} + 11 a + 6\right)\cdot 19 + \left(15 a^{4} + 7 a^{3} + 11 a^{2} + 17 a + 6\right)\cdot 19^{2} + \left(6 a^{4} + 17 a^{2} + 5 a + 10\right)\cdot 19^{3} + \left(13 a^{4} + 11 a^{3} + 13 a^{2} + 4 a + 10\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 12 }$ | $=$ | \( 13 a^{4} + 13 a^{3} + 13 a^{2} + 16 a + \left(9 a^{4} + 11 a^{3} + 13 a^{2} + 16 a + 12\right)\cdot 19 + \left(6 a^{4} + 10 a^{3} + 9 a^{2} + 3 a + 6\right)\cdot 19^{2} + \left(3 a^{4} + 10 a^{3} + a^{2} + 3 a + 15\right)\cdot 19^{3} + \left(16 a^{4} + a^{3} + 5 a^{2} + 17 a + 2\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 13 }$ | $=$ | \( 15 a^{4} + 16 a^{3} + a^{2} + 17 a + 12 + \left(4 a^{4} + 11 a^{3} + 13 a + 1\right)\cdot 19 + \left(16 a^{2} + 18 a + 2\right)\cdot 19^{2} + \left(13 a^{4} + 2 a^{2} + 6 a + 5\right)\cdot 19^{3} + \left(8 a^{4} + 8 a^{3} + 2 a^{2} + 14 a + 17\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 14 }$ | $=$ | \( 16 a^{4} + 18 a^{3} + 14 a^{2} + 9 a + 12 + \left(12 a^{4} + 8 a^{3} + 16 a^{2} + 4 a + 5\right)\cdot 19 + \left(3 a^{4} + 15 a^{3} + 8 a^{2} + 4 a + 14\right)\cdot 19^{2} + \left(17 a^{4} + 15 a^{3} + 4 a^{2} + 2 a + 13\right)\cdot 19^{3} + \left(2 a^{4} + 18 a^{3} + 14 a^{2} + 12 a + 6\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 15 }$ | $=$ | \( 18 a^{4} + 16 a^{3} + 10 a^{2} + 16 a + 2 + \left(6 a^{4} + 9 a^{3} + 4 a^{2} + 11 a + 15\right)\cdot 19 + \left(16 a^{4} + 8 a^{3} + 8 a^{2} + 18 a + 7\right)\cdot 19^{2} + \left(11 a^{4} + 17 a^{3} + 18 a + 16\right)\cdot 19^{3} + \left(10 a^{4} + 13 a^{3} + 10 a^{2} + 4 a + 7\right)\cdot 19^{4} +O(19^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$15$ | $2$ | $(2,14)(3,7)(4,11)(5,8)(6,12)(9,15)(10,13)$ | $0$ |
$2$ | $3$ | $(1,14,2)(3,11,6)(4,7,12)(5,15,10)(8,13,9)$ | $-1$ |
$2$ | $5$ | $(1,13,3,7,10)(2,8,6,4,15)(5,14,9,11,12)$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ |
$2$ | $5$ | $(1,3,10,13,7)(2,6,15,8,4)(5,9,12,14,11)$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ |
$2$ | $15$ | $(1,9,6,7,5,2,13,11,4,10,14,8,3,12,15)$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ |
$2$ | $15$ | $(1,6,5,13,4,14,3,15,9,7,2,11,10,8,12)$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ |
$2$ | $15$ | $(1,5,4,3,9,2,10,12,6,13,14,15,7,11,8)$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ |
$2$ | $15$ | $(1,11,15,13,12,2,3,5,8,7,14,6,10,9,4)$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.