Basic invariants
Dimension: | $2$ |
Group: | $D_{15}$ |
Conductor: | \(671\)\(\medspace = 11 \cdot 61 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 15.1.61243167054566186591.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $D_{15}$ |
Parity: | odd |
Determinant: | 1.671.2t1.a.a |
Projective image: | $D_{15}$ |
Projective stem field: | Galois closure of 15.1.61243167054566186591.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{15} - 3 x^{14} + 6 x^{13} - x^{12} - 20 x^{11} + 56 x^{10} - 81 x^{9} + 46 x^{8} + 90 x^{7} + \cdots + 11 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{5} + 3x + 27 \)
Roots:
$r_{ 1 }$ | $=$ | \( 6 a^{4} + 4 a^{3} + 5 a^{2} + 27 a + 3 + \left(5 a^{4} + 15 a^{3} + 9 a^{2} + 10 a + 24\right)\cdot 29 + \left(14 a^{4} + 8 a^{3} + 26 a^{2} + 25 a + 10\right)\cdot 29^{2} + \left(24 a^{4} + 18 a^{3} + 11 a^{2} + 19 a + 12\right)\cdot 29^{3} + \left(15 a^{4} + 6 a^{3} + 20 a^{2} + 19 a + 26\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 2 }$ | $=$ | \( 9 a^{4} + 19 a^{3} + 19 a^{2} + 25 a + 16 + \left(3 a^{4} + 6 a^{3} + 20 a^{2} + 15 a + 19\right)\cdot 29 + \left(15 a^{4} + 5 a^{3} + 20 a^{2} + 23 a + 24\right)\cdot 29^{2} + \left(9 a^{4} + 22 a^{3} + 25 a^{2} + 26 a + 22\right)\cdot 29^{3} + \left(9 a^{4} + 4 a^{3} + 14 a^{2} + 23 a + 10\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 3 }$ | $=$ | \( 9 a^{4} + 25 a^{3} + 21 a^{2} + 2 a + 16 + \left(3 a^{4} + 26 a^{3} + 19 a^{2} + 27 a + 19\right)\cdot 29 + \left(16 a^{3} + 13 a^{2} + 13 a + 17\right)\cdot 29^{2} + \left(16 a^{4} + 20 a^{3} + 16 a + 3\right)\cdot 29^{3} + \left(23 a^{4} + a^{3} + 2 a^{2} + 12 a + 10\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 4 }$ | $=$ | \( 11 a^{4} + 25 a^{2} + 3 a + 15 + \left(3 a^{4} + 9 a^{3} + 16 a^{2} + 19 a + 25\right)\cdot 29 + \left(10 a^{4} + 4 a^{3} + 7 a^{2} + 9 a + 6\right)\cdot 29^{2} + \left(9 a^{4} + 7 a^{3} + 22 a^{2} + 4 a + 28\right)\cdot 29^{3} + \left(9 a^{4} + 20 a^{3} + 12 a^{2} + 4 a + 4\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 5 }$ | $=$ | \( 13 a^{4} + 15 a^{3} + a^{2} + 23 a + 14 + \left(13 a^{4} + a^{3} + 27 a^{2} + 14 a + 26\right)\cdot 29 + \left(6 a^{4} + 8 a^{3} + 14 a^{2} + 22 a + 3\right)\cdot 29^{2} + \left(24 a^{4} + 28 a^{3} + 27 a^{2} + 13 a\right)\cdot 29^{3} + \left(16 a^{4} + 18 a^{3} + 20 a^{2} + 23\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 6 }$ | $=$ | \( 14 a^{4} + 8 a^{3} + 2 a^{2} + 21 a + 28 + \left(13 a^{4} + 25 a^{3} + 14 a^{2} + 8 a + 14\right)\cdot 29 + \left(20 a^{3} + 7 a^{2} + 20 a + 18\right)\cdot 29^{2} + \left(24 a^{4} + a^{3} + 2 a + 28\right)\cdot 29^{3} + \left(28 a^{4} + 10 a^{3} + 15 a^{2} + 15 a + 16\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 7 }$ | $=$ | \( 16 a^{4} + 25 a^{3} + 20 a^{2} + 20 a + 27 + \left(19 a^{4} + 20 a^{3} + 25 a^{2} + 10 a + 17\right)\cdot 29 + \left(22 a^{4} + 6 a^{3} + 8 a^{2} + 27 a + 19\right)\cdot 29^{2} + \left(15 a^{4} + 13 a^{3} + 24 a^{2} + 8 a + 14\right)\cdot 29^{3} + \left(3 a^{4} + 18 a^{3} + 3 a^{2} + 19 a + 8\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 8 }$ | $=$ | \( 18 a^{4} + 24 a^{3} + 23 a^{2} + 26 a + 26 + \left(4 a^{4} + 7 a^{3} + 4 a^{2} + 22 a + 16\right)\cdot 29 + \left(24 a^{4} + 20 a^{3} + 9 a^{2} + 9 a + 11\right)\cdot 29^{2} + \left(8 a^{4} + 15 a^{3} + 13 a^{2} + 8 a + 15\right)\cdot 29^{3} + \left(10 a^{4} + 20 a^{3} + 13 a^{2} + 19 a + 1\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 9 }$ | $=$ | \( 20 a^{4} + 13 a^{3} + 26 a^{2} + 20 a + 25 + \left(26 a^{4} + 19 a^{3} + 10 a^{2} + 3 a + 11\right)\cdot 29 + \left(24 a^{4} + 10 a^{3} + 8 a^{2} + 3 a + 13\right)\cdot 29^{2} + \left(10 a^{4} + 2 a^{3} + 14 a^{2} + 23 a + 14\right)\cdot 29^{3} + \left(5 a^{4} + 22 a^{3} + 10 a^{2} + a + 24\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 10 }$ | $=$ | \( 21 a^{4} + 27 a^{3} + 20 a^{2} + 17 a + 10 + \left(12 a^{4} + 14 a^{3} + 6 a^{2} + 5 a + 7\right)\cdot 29 + \left(9 a^{4} + 7 a^{3} + 21 a^{2} + 4 a + 5\right)\cdot 29^{2} + \left(5 a^{4} + 5 a^{3} + 21 a^{2} + 12 a + 1\right)\cdot 29^{3} + \left(17 a^{4} + 17 a^{3} + 26 a^{2} + 10 a + 18\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 11 }$ | $=$ | \( 22 a^{4} + 19 a^{3} + 28 a^{2} + 19 a + 24 + \left(14 a^{4} + 11 a^{3} + 28 a^{2} + 11 a + 23\right)\cdot 29 + \left(13 a^{4} + 7 a^{3} + 23 a^{2} + 13 a + 20\right)\cdot 29^{2} + \left(17 a^{4} + a^{3} + 18 a^{2} + a + 18\right)\cdot 29^{3} + \left(3 a^{4} + 10 a^{3} + 10 a^{2} + 12 a + 8\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 12 }$ | $=$ | \( 24 a^{4} + 17 a^{3} + 27 a^{2} + 10 a + 23 + \left(24 a^{4} + 3 a^{3} + 20 a^{2} + 27 a + 24\right)\cdot 29 + \left(9 a^{4} + 20 a^{3} + 10 a^{2} + 8 a + 17\right)\cdot 29^{2} + \left(18 a^{4} + a^{3} + 27 a^{2} + 11 a + 26\right)\cdot 29^{3} + \left(18 a^{4} + 10 a^{3} + 4 a^{2} + 6 a + 9\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 13 }$ | $=$ | \( 25 a^{4} + 6 a^{3} + 12 a^{2} + 15 a + 8 + \left(9 a^{4} + 20 a^{3} + 27 a^{2} + 26 a + 6\right)\cdot 29 + \left(21 a^{4} + 14 a^{3} + 17 a^{2} + 12 a + 22\right)\cdot 29^{2} + \left(4 a^{4} + 4 a^{3} + 9 a^{2} + 8 a + 28\right)\cdot 29^{3} + \left(16 a^{4} + 5 a^{3} + 3 a + 3\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 14 }$ | $=$ | \( 26 a^{4} + 16 a^{3} + 18 a^{2} + 2 a + 22 + \left(6 a^{4} + 22 a^{3} + 26 a^{2} + 11 a + 10\right)\cdot 29 + \left(6 a^{4} + 17 a^{3} + 25 a^{2} + 2 a + 3\right)\cdot 29^{2} + \left(20 a^{4} + 19 a^{3} + 17 a^{2} + 18 a + 2\right)\cdot 29^{3} + \left(12 a^{4} + 14 a^{3} + 13 a^{2} + 16 a + 13\right)\cdot 29^{4} +O(29^{5})\) |
$r_{ 15 }$ | $=$ | \( 27 a^{4} + 14 a^{3} + 14 a^{2} + 2 a + 7 + \left(11 a^{4} + 26 a^{3} + a^{2} + 16 a + 11\right)\cdot 29 + \left(24 a^{4} + 4 a^{3} + 15 a^{2} + 5 a + 6\right)\cdot 29^{2} + \left(22 a^{4} + 12 a^{3} + 25 a^{2} + 27 a + 14\right)\cdot 29^{3} + \left(11 a^{4} + 22 a^{3} + 3 a^{2} + 8 a + 22\right)\cdot 29^{4} +O(29^{5})\) |
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 |
$1$ | $1$ | $()$ | $2$ |
$15$ | $2$ | $(2,13)(3,10)(4,7)(5,6)(8,14)(9,11)(12,15)$ | $0$ |
$2$ | $3$ | $(1,6,5)(2,9,15)(3,7,8)(4,10,14)(11,13,12)$ | $-1$ |
$2$ | $5$ | $(1,3,12,15,10)(2,14,6,7,11)(4,5,8,13,9)$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ |
$2$ | $5$ | $(1,12,10,3,15)(2,6,11,14,7)(4,8,9,5,13)$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ |
$2$ | $15$ | $(1,7,13,15,14,5,3,11,9,10,6,8,12,2,4)$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ |
$2$ | $15$ | $(1,13,14,3,9,6,12,4,7,15,5,11,10,8,2)$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ |
$2$ | $15$ | $(1,14,9,12,7,5,10,2,13,3,6,4,15,11,8)$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ |
$2$ | $15$ | $(1,11,4,3,2,5,12,14,8,15,6,13,10,7,9)$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ |
The blue line marks the conjugacy class containing complex conjugation.