Basic invariants
Dimension: | $16$ |
Group: | $((C_3^2:Q_8):C_3):C_2$ |
Conductor: | \(392\!\cdots\!625\)\(\medspace = 3^{18} \cdot 5^{8} \cdot 11^{10} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 9.3.21793521976875.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | 24T1334 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $C_3^2:\GL(2,3)$ |
Projective stem field: | Galois closure of 9.3.21793521976875.2 |
Defining polynomial
$f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 16x^{6} - 33x^{5} - 36x^{4} + 120x^{3} + 99x^{2} - 297x + 77 \)
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The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{4} + 2x^{2} + 11x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 3 a^{3} + 16 a^{2} + 5 a + 18 + \left(14 a^{3} + 18 a^{2} + 12 a + 11\right)\cdot 19 + \left(5 a^{3} + 8 a^{2} + 8 a\right)\cdot 19^{2} + \left(11 a^{3} + 5 a^{2} + 11 a + 18\right)\cdot 19^{3} + \left(3 a^{3} + 15 a^{2} + 14 a + 8\right)\cdot 19^{4} + \left(13 a^{3} + 5 a^{2} + 2 a + 17\right)\cdot 19^{5} + \left(17 a^{3} + 5 a^{2} + 8 a + 8\right)\cdot 19^{6} + \left(17 a^{3} + 9 a^{2} + 12 a + 17\right)\cdot 19^{7} + \left(11 a^{3} + 11 a^{2} + 16 a + 5\right)\cdot 19^{8} + \left(5 a^{2} + 16 a + 10\right)\cdot 19^{9} +O(19^{10})\)
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$r_{ 2 }$ | $=$ |
\( 10 a^{3} + 6 a^{2} + 8 a + 9 + \left(10 a^{3} + 16 a^{2} + 11 a + 10\right)\cdot 19 + \left(2 a^{3} + 6 a^{2} + 5 a + 16\right)\cdot 19^{2} + \left(16 a^{3} + 16 a + 13\right)\cdot 19^{3} + \left(14 a^{3} + 13 a^{2} + a + 17\right)\cdot 19^{4} + \left(6 a^{3} + 2 a^{2} + a + 3\right)\cdot 19^{5} + \left(16 a^{3} + 13 a^{2} + 18 a\right)\cdot 19^{6} + \left(a^{3} + 13 a^{2} + 14 a + 13\right)\cdot 19^{7} + \left(5 a^{3} + 4 a^{2} + 15 a + 11\right)\cdot 19^{8} + \left(11 a^{3} + 7 a^{2} + 18 a + 7\right)\cdot 19^{9} +O(19^{10})\)
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$r_{ 3 }$ | $=$ |
\( 13 + 10\cdot 19 + 2\cdot 19^{2} + 3\cdot 19^{3} + 7\cdot 19^{4} + 18\cdot 19^{5} + 19^{6} + 15\cdot 19^{7} + 5\cdot 19^{8} + 11\cdot 19^{9} +O(19^{10})\)
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$r_{ 4 }$ | $=$ |
\( 17 a^{3} + 7 a^{2} + 6 a + 1 + \left(6 a^{3} + 8 a^{2} + 13 a + 8\right)\cdot 19 + \left(6 a^{3} + 11 a^{2} + 11 a + 3\right)\cdot 19^{2} + \left(6 a^{3} + 11 a^{2} + 8 a + 2\right)\cdot 19^{3} + \left(3 a^{3} + 16 a^{2} + 15 a + 8\right)\cdot 19^{4} + \left(17 a^{3} + 12 a^{2} + 16 a\right)\cdot 19^{5} + \left(2 a^{3} + 2 a^{2} + 12 a + 3\right)\cdot 19^{6} + \left(3 a^{3} + 18 a^{2} + 2 a + 4\right)\cdot 19^{7} + \left(2 a^{3} + 2 a^{2} + 2\right)\cdot 19^{8} + \left(2 a^{3} + 14 a + 17\right)\cdot 19^{9} +O(19^{10})\)
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$r_{ 5 }$ | $=$ |
\( 2 a^{3} + 4 a^{2} + 10 a + 17 + \left(13 a^{3} + 2 a^{2} + 18 a + 12\right)\cdot 19 + \left(13 a^{3} + 7 a^{2} + 12 a + 13\right)\cdot 19^{2} + \left(18 a^{3} + 6 a + 6\right)\cdot 19^{3} + \left(13 a^{3} + 2\right)\cdot 19^{4} + \left(14 a^{3} + 4 a^{2} + 12 a + 9\right)\cdot 19^{5} + \left(4 a^{3} + 6 a^{2} + 11 a + 16\right)\cdot 19^{6} + \left(4 a^{3} + 5 a^{2} + 6 a + 14\right)\cdot 19^{7} + \left(11 a^{3} + 6 a^{2} + 15 a + 6\right)\cdot 19^{8} + \left(4 a + 12\right)\cdot 19^{9} +O(19^{10})\)
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$r_{ 6 }$ | $=$ |
\( 15 a^{3} + 12 a^{2} + 18 a + 4 + \left(17 a^{3} + 16 a^{2} + 16 a + 4\right)\cdot 19 + \left(17 a^{3} + 9 a^{2} + 4 a + 18\right)\cdot 19^{2} + \left(15 a^{3} + 5 a^{2} + 12 a + 7\right)\cdot 19^{3} + \left(8 a^{2} + 5 a + 11\right)\cdot 19^{4} + \left(12 a^{3} + 18 a^{2} + 8 a + 15\right)\cdot 19^{5} + \left(15 a^{3} + 18 a^{2} + 17 a + 4\right)\cdot 19^{6} + \left(17 a^{3} + a^{2} + 4 a + 14\right)\cdot 19^{7} + \left(10 a^{3} + 2 a^{2} + 12 a + 9\right)\cdot 19^{8} + \left(16 a^{3} + a^{2} + 13 a + 2\right)\cdot 19^{9} +O(19^{10})\)
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$r_{ 7 }$ | $=$ |
\( 11 a^{3} + 16 a^{2} + 2 a + 13 + \left(15 a^{3} + 2 a^{2} + 10 a\right)\cdot 19 + \left(3 a^{3} + 14 a^{2} + 14 a + 1\right)\cdot 19^{2} + \left(6 a^{3} + 12 a^{2} + 2 a + 6\right)\cdot 19^{3} + \left(8 a^{3} + 16 a^{2} + 11 a + 10\right)\cdot 19^{4} + \left(4 a^{3} + 12 a^{2} + 16 a + 18\right)\cdot 19^{5} + \left(a^{3} + 18 a^{2} + 9 a + 13\right)\cdot 19^{6} + \left(14 a^{3} + 16 a^{2} + 11 a + 7\right)\cdot 19^{7} + \left(10 a^{3} + 5 a^{2} + 13 a + 2\right)\cdot 19^{8} + \left(9 a^{3} + 10 a^{2} + 6\right)\cdot 19^{9} +O(19^{10})\)
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$r_{ 8 }$ | $=$ |
\( 4 a^{2} + a + 5 + \left(3 a^{3} + a^{2} + 16\right)\cdot 19 + \left(3 a^{3} + a^{2} + 12 a + 18\right)\cdot 19^{2} + \left(17 a^{3} + 6 a^{2} + 14 a + 9\right)\cdot 19^{3} + \left(8 a^{3} + 3 a^{2} + 4 a + 7\right)\cdot 19^{4} + \left(4 a^{3} + 16 a^{2} + 3 a + 3\right)\cdot 19^{5} + \left(17 a^{3} + 6 a^{2} + 9 a + 16\right)\cdot 19^{6} + \left(14 a^{3} + 13 a^{2} + 12 a + 10\right)\cdot 19^{7} + \left(3 a^{3} + 15 a^{2} + 2 a + 4\right)\cdot 19^{8} + \left(6 a^{3} + 12 a^{2} + 5 a + 11\right)\cdot 19^{9} +O(19^{10})\)
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$r_{ 9 }$ | $=$ |
\( 18 a^{3} + 11 a^{2} + 7 a + 18 + \left(13 a^{3} + 9 a^{2} + 12 a\right)\cdot 19 + \left(3 a^{3} + 16 a^{2} + 5 a + 1\right)\cdot 19^{2} + \left(3 a^{3} + 14 a^{2} + 3 a + 8\right)\cdot 19^{3} + \left(3 a^{3} + 2 a^{2} + 3 a + 2\right)\cdot 19^{4} + \left(3 a^{3} + 3 a^{2} + 15 a + 8\right)\cdot 19^{5} + \left(4 a^{2} + 7 a + 10\right)\cdot 19^{6} + \left(2 a^{3} + 16 a^{2} + 10 a + 16\right)\cdot 19^{7} + \left(a^{3} + 7 a^{2} + 18 a + 7\right)\cdot 19^{8} + \left(10 a^{3} + a + 16\right)\cdot 19^{9} +O(19^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
$1$ | $1$ | $()$ | $16$ |
$9$ | $2$ | $(2,8)(3,4)(5,7)(6,9)$ | $0$ |
$36$ | $2$ | $(1,6)(2,5)(4,7)$ | $0$ |
$8$ | $3$ | $(1,3,4)(2,7,9)(5,8,6)$ | $-2$ |
$24$ | $3$ | $(2,4,6)(3,9,8)$ | $-2$ |
$48$ | $3$ | $(1,3,2)(4,5,8)(6,7,9)$ | $1$ |
$54$ | $4$ | $(2,3,8,4)(5,9,7,6)$ | $0$ |
$72$ | $6$ | $(1,5,2,4,8,9)(3,6,7)$ | $0$ |
$72$ | $6$ | $(1,7,8,6,2,3)(4,5)$ | $0$ |
$54$ | $8$ | $(1,7,9,5,2,4,3,6)$ | $0$ |
$54$ | $8$ | $(1,4,9,6,2,7,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.