Basic invariants
Dimension: | $2$ |
Group: | $D_{15}$ |
Conductor: | \(3639\)\(\medspace = 3 \cdot 1213 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 15.1.8450344007000266933623879.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $D_{15}$ |
Parity: | odd |
Projective image: | $D_{15}$ |
Projective field: | Galois closure of 15.1.8450344007000266933623879.1 |
Galois action
Roots of defining polynomial
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:
$r_{ 1 }$ | $=$ | \( a^{4} + 4 a^{3} + 3 a^{2} + 2 a + 11 + \left(13 a^{4} + 16 a^{3} + 7 a^{2} + 2 a + 7\right)\cdot 19 + \left(15 a^{4} + 9 a^{3} + 17 a^{2} + 11 a + 17\right)\cdot 19^{2} + \left(4 a^{4} + 7 a^{3} + 10 a^{2} + 6 a + 7\right)\cdot 19^{3} + \left(18 a^{4} + 3 a^{3} + 8 a^{2} + 11 a + 2\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 2 }$ | $=$ | \( 2 a^{4} + 11 a^{3} + 16 a^{2} + 7 a + 4 + \left(8 a^{4} + 16 a^{3} + 5 a^{2} + 6 a\right)\cdot 19 + \left(2 a^{4} + 13 a^{3} + 8 a^{2} + 8 a + 15\right)\cdot 19^{2} + \left(3 a^{4} + 14 a^{3} + 5 a^{2} + 14 a + 13\right)\cdot 19^{3} + \left(13 a^{4} + 3 a^{3} + 16 a^{2} + 3 a + 16\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 3 }$ | $=$ | \( 2 a^{4} + 16 a^{3} + 2 a^{2} + 7 a + 4 + \left(17 a^{4} + 18 a^{3} + 4 a^{2} + 18 a + 17\right)\cdot 19 + \left(14 a^{4} + 10 a^{3} + 18 a^{2} + 15 a + 7\right)\cdot 19^{2} + \left(17 a^{4} + 11 a^{3} + 18 a^{2} + 16 a + 15\right)\cdot 19^{3} + \left(17 a^{4} + 10 a^{3} + 5 a^{2} + 14 a + 16\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 4 }$ | $=$ | \( 4 a^{4} + 4 a^{3} + 13 a^{2} + 5 a + 12 + \left(a^{4} + 7 a^{3} + 16 a^{2} + 15 a + 10\right)\cdot 19 + \left(12 a^{4} + 5 a^{3} + 13 a^{2} + 11 a + 15\right)\cdot 19^{2} + \left(14 a^{4} + 13 a^{3} + a^{2} + 14 a + 2\right)\cdot 19^{3} + \left(15 a^{4} + 9 a^{3} + 4 a^{2} + 18 a + 8\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 5 }$ | $=$ | \( 4 a^{4} + 13 a^{3} + 8 a + 18 + \left(15 a^{4} + 8 a^{3} + 3 a^{2} + 17 a\right)\cdot 19 + \left(4 a^{4} + 4 a + 6\right)\cdot 19^{2} + \left(11 a^{4} + 5 a^{3} + 16 a^{2} + 5 a + 13\right)\cdot 19^{3} + \left(5 a^{4} + 14 a^{3} + 3 a^{2} + 6 a + 18\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 6 }$ | $=$ | \( 5 a^{4} + 15 a^{3} + 12 a^{2} + 2 a + 3 + \left(2 a^{4} + 2 a^{3} + 5 a^{2} + 12 a + 6\right)\cdot 19 + \left(14 a^{4} + 9 a^{3} + 5 a^{2} + 7 a + 5\right)\cdot 19^{2} + \left(8 a^{4} + 18 a^{3} + 9 a^{2} + 2 a + 3\right)\cdot 19^{3} + \left(13 a^{4} + 15 a^{3} + 17 a^{2} + 15 a + 12\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 7 }$ | $=$ | \( 6 a^{4} + 7 a^{3} + 10 a^{2} + 16 a + 7 + \left(9 a^{4} + 11 a^{3} + 11 a^{2} + 15 a + 15\right)\cdot 19 + \left(11 a^{4} + 6 a^{3} + 16 a^{2} + 13 a + 13\right)\cdot 19^{2} + \left(10 a^{4} + 12 a^{3} + 6 a + 10\right)\cdot 19^{3} + \left(10 a^{4} + 3 a^{3} + 11 a^{2} + 4 a\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 8 }$ | $=$ | \( 10 a^{4} + 12 a^{3} + 4 a^{2} + 2 a + 9 + \left(12 a^{4} + 4 a^{3} + a^{2} + 13 a + 5\right)\cdot 19 + \left(16 a^{4} + 12 a^{3} + 14 a^{2} + 2\right)\cdot 19^{2} + \left(8 a^{4} + 14 a^{3} + 18 a + 5\right)\cdot 19^{3} + \left(13 a^{4} + 7 a^{3} + 5 a^{2} + 3 a + 2\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 9 }$ | $=$ | \( 11 a^{4} + 11 a^{3} + 3 a^{2} + 7 a + 8 + \left(16 a^{4} + 6 a^{3} + 11 a^{2} + 2 a + 6\right)\cdot 19 + \left(13 a^{4} + 15 a^{3} + 12 a^{2} + 11 a + 4\right)\cdot 19^{2} + \left(4 a^{4} + 14 a^{3} + 18 a^{2} + 14 a + 6\right)\cdot 19^{3} + \left(8 a^{4} + 16 a^{3} + a^{2} + 15 a + 10\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 10 }$ | $=$ | \( 12 a^{4} + 11 a^{3} + 13 a^{2} + 5 a + 12 + \left(13 a^{4} + 8 a^{3} + 6 a^{2} + 9 a + 13\right)\cdot 19 + \left(12 a^{4} + 6 a^{3} + 3 a^{2} + 18\right)\cdot 19^{2} + \left(2 a^{4} + 6 a^{3} + 12 a^{2} + 9 a + 16\right)\cdot 19^{3} + \left(6 a^{3} + 3 a^{2} + 15 a + 15\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 11 }$ | $=$ | \( 13 a^{4} + a^{3} + 5 a^{2} + 8 a + 2 + \left(16 a^{4} + 5 a^{2} + 6 a + 3\right)\cdot 19 + \left(18 a^{4} + 16 a^{3} + 9 a^{2} + 13 a + 11\right)\cdot 19^{2} + \left(6 a^{4} + 13 a^{3} + 13 a^{2} + 9 a + 16\right)\cdot 19^{3} + \left(11 a^{4} + 4 a^{3} + a^{2} + 11 a + 12\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 12 }$ | $=$ | \( 15 a^{4} + 10 a^{3} + 15 a^{2} + a + 18 + \left(a^{3} + 9 a^{2} + 11 a + 8\right)\cdot 19 + \left(16 a^{4} + 14 a^{2} + 8 a + 12\right)\cdot 19^{2} + \left(5 a^{3} + 17 a^{2} + 14 a + 4\right)\cdot 19^{3} + \left(7 a^{4} + 8 a^{3} + 14 a^{2} + 5 a + 11\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 13 }$ | $=$ | \( 15 a^{4} + 12 a^{3} + 18 a^{2} + 12 a + 10 + \left(17 a^{4} + 5 a^{3} + 6 a^{2} + 7\right)\cdot 19 + \left(12 a^{4} + 13 a^{3} + 8 a^{2} + 11 a + 6\right)\cdot 19^{2} + \left(12 a^{4} + 5 a^{3} + 9 a^{2} + 7 a + 1\right)\cdot 19^{3} + \left(6 a^{4} + 8 a^{3} + 17 a^{2} + 8 a + 13\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 14 }$ | $=$ | \( 15 a^{4} + 16 a^{3} + 11 a^{2} + 18 a + 18 + \left(10 a^{4} + 12 a^{3} + a^{2} + 5 a + 10\right)\cdot 19 + \left(11 a^{4} + 7 a^{3} + 2 a^{2} + 12 a + 13\right)\cdot 19^{2} + \left(a^{4} + 12 a^{3} + 13 a^{2} + 15 a + 7\right)\cdot 19^{3} + \left(3 a^{4} + 5 a^{3} + 15 a^{2} + 13 a + 14\right)\cdot 19^{4} +O(19^{5})\) |
$r_{ 15 }$ | $=$ | \( 18 a^{4} + 9 a^{3} + 8 a^{2} + 14 a + 3 + \left(15 a^{4} + 11 a^{3} + 17 a^{2} + 15 a\right)\cdot 19 + \left(11 a^{4} + 5 a^{3} + 7 a^{2} + a + 2\right)\cdot 19^{2} + \left(4 a^{4} + 15 a^{3} + 3 a^{2} + 15 a + 7\right)\cdot 19^{3} + \left(7 a^{4} + 13 a^{3} + 5 a^{2} + 2 a + 15\right)\cdot 19^{4} +O(19^{5})\) |
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 values | |||
$c1$ | $c2$ | $c3$ | $c4$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ | $2$ | $2$ |
$15$ | $2$ | $(1,12)(2,13)(3,8)(4,15)(5,6)(7,10)(11,14)$ | $0$ | $0$ | $0$ | $0$ |
$2$ | $3$ | $(1,2,5)(3,9,8)(4,7,11)(6,13,12)(10,15,14)$ | $-1$ | $-1$ | $-1$ | $-1$ |
$2$ | $5$ | $(1,15,11,13,8)(2,14,4,12,3)(5,10,7,6,9)$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ |
$2$ | $5$ | $(1,11,8,15,13)(2,4,3,14,12)(5,7,9,10,6)$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ | $-\zeta_{15}^{7} + \zeta_{15}^{3} - \zeta_{15}^{2}$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ | $\zeta_{15}^{7} - \zeta_{15}^{3} + \zeta_{15}^{2} - 1$ |
$2$ | $15$ | $(1,14,7,13,3,5,15,4,6,8,2,10,11,12,9)$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ |
$2$ | $15$ | $(1,7,3,15,6,2,11,9,14,13,5,4,8,10,12)$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ |
$2$ | $15$ | $(1,3,6,11,14,5,8,12,7,15,2,9,13,4,10)$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ |
$2$ | $15$ | $(1,4,9,15,12,5,11,3,10,13,2,7,8,14,6)$ | $2 \zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} + \zeta_{15}^{2} - \zeta_{15} + 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + \zeta_{15} + 1$ | $-\zeta_{15}^{6} + \zeta_{15}^{4} - \zeta_{15}$ |