Basic invariants
| Dimension: | $16$ |
| Group: | $((C_3^2:Q_8):C_3):C_2$ |
| Conductor: | \(166\!\cdots\!936\)\(\medspace = 2^{10} \cdot 3^{18} \cdot 29^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.3.749306075343552.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 24T1334 |
| Parity: | even |
| Projective image: | $C_3^2:\GL(2,3)$ |
| Projective field: | Galois closure of 9.3.749306075343552.2 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$:
\( x^{4} + 3x^{2} + 12x + 2 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a^{3} + a^{2} + 5 a + 8 + \left(7 a^{3} + 10 a^{2} + 4 a + 9\right)\cdot 13 + \left(6 a^{3} + 6 a^{2} + 8 a + 10\right)\cdot 13^{2} + \left(5 a^{3} + 7 a^{2} + 6 a + 7\right)\cdot 13^{3} + \left(9 a^{3} + a + 6\right)\cdot 13^{4} + \left(10 a^{3} + a^{2} + 9 a + 1\right)\cdot 13^{5} + \left(12 a^{3} + 12 a^{2} + 6 a + 11\right)\cdot 13^{6} + \left(7 a^{3} + 2 a^{2} + 10 a + 11\right)\cdot 13^{7} + \left(a^{3} + 2 a^{2} + 8 a + 10\right)\cdot 13^{8} + \left(9 a^{3} + 9 a^{2} + 9 a + 5\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a^{3} + 10 a^{2} + 3 a + 2 + \left(6 a^{3} + 8 a^{2} + 3 a\right)\cdot 13 + \left(4 a^{3} + 5 a^{2}\right)\cdot 13^{2} + \left(9 a^{3} + 12 a^{2} + a + 10\right)\cdot 13^{3} + \left(8 a^{3} + a^{2} + 4 a + 10\right)\cdot 13^{4} + \left(2 a^{3} + 4 a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(6 a^{3} + a^{2} + 3 a + 8\right)\cdot 13^{6} + \left(2 a^{3} + 12 a + 8\right)\cdot 13^{7} + \left(9 a^{3} + 10 a^{2} + 8 a + 2\right)\cdot 13^{8} + \left(4 a^{3} + 7 a^{2} + 10 a + 10\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a^{3} + 4 a^{2} + 6 a + 1 + \left(12 a^{3} + 12 a^{2} + 3 a + 11\right)\cdot 13 + \left(9 a^{2} + 12 a + 3\right)\cdot 13^{2} + \left(7 a^{3} + 12 a^{2} + 10 a + 10\right)\cdot 13^{3} + \left(7 a^{3} + 5 a^{2} + 8 a + 10\right)\cdot 13^{4} + \left(4 a^{3} + 3 a^{2} + a + 1\right)\cdot 13^{5} + \left(5 a^{3} + 4 a^{2} + 10\right)\cdot 13^{6} + \left(8 a^{3} + 12 a + 11\right)\cdot 13^{7} + \left(2 a^{2} + 1\right)\cdot 13^{8} + \left(2 a^{2} + 7 a + 11\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 12 + 7\cdot 13 + 3\cdot 13^{2} + 11\cdot 13^{3} + 4\cdot 13^{5} + 10\cdot 13^{6} + 5\cdot 13^{8} + 2\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 11 a^{3} + 8 a^{2} + 2 a + 3 + \left(12 a^{3} + 12 a^{2} + 11 a + 7\right)\cdot 13 + \left(8 a^{3} + 8 a^{2} + a + 12\right)\cdot 13^{2} + \left(4 a^{3} + 2 a^{2} + 12 a + 11\right)\cdot 13^{3} + \left(10 a^{3} + 4 a^{2} + 2\right)\cdot 13^{4} + \left(5 a^{2} + 3 a\right)\cdot 13^{5} + \left(10 a^{3} + 2 a^{2} + 2 a + 6\right)\cdot 13^{6} + \left(4 a^{3} + 5 a^{2} + 4\right)\cdot 13^{7} + \left(a^{3} + 8 a^{2} + 4 a + 1\right)\cdot 13^{8} + \left(11 a^{3} + 5 a^{2} + 11 a + 6\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 6 a^{3} + a^{2} + 10 a + 2 + \left(12 a^{3} + 11 a^{2} + 10 a + 7\right)\cdot 13 + \left(8 a^{3} + 9 a^{2} + 10 a + 10\right)\cdot 13^{2} + \left(11 a^{3} + a^{2} + 6 a + 2\right)\cdot 13^{3} + \left(3 a^{3} + 8 a^{2} + 10 a + 7\right)\cdot 13^{4} + \left(10 a^{3} + 10 a^{2} + 6 a + 5\right)\cdot 13^{5} + \left(a^{3} + 9 a + 5\right)\cdot 13^{6} + \left(11 a^{3} + 7 a^{2} + 12 a + 7\right)\cdot 13^{7} + \left(4 a^{3} + 6 a^{2} + 2 a + 7\right)\cdot 13^{8} + \left(3 a^{3} + 2 a^{2} + 4 a + 2\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 7 a^{2} + 5 a + 9 + \left(6 a^{3} + 5 a^{2} + 7 a + 5\right)\cdot 13 + \left(9 a^{3} + 12 a^{2} + 7 a + 12\right)\cdot 13^{2} + \left(a^{3} + 3 a^{2} + a\right)\cdot 13^{3} + \left(5 a^{3} + 11 a^{2} + 5 a + 10\right)\cdot 13^{4} + \left(10 a^{2} + 8 a + 7\right)\cdot 13^{5} + \left(6 a^{3} + 8 a^{2} + 9 a + 3\right)\cdot 13^{6} + \left(11 a^{3} + 2 a^{2} + 3 a + 10\right)\cdot 13^{7} + \left(5 a^{3} + 2 a^{2} + 3\right)\cdot 13^{8} + \left(12 a^{2} + 5 a + 10\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 7 a^{3} + 11 a^{2} + 12 a + 4 + \left(2 a^{3} + 8 a^{2} + 11 a + 6\right)\cdot 13 + \left(2 a^{3} + 8 a^{2} + 4 a + 9\right)\cdot 13^{2} + \left(7 a^{3} + 7 a^{2} + 2\right)\cdot 13^{3} + \left(10 a^{2} + 5 a + 2\right)\cdot 13^{4} + \left(12 a^{3} + 11 a^{2} + 2 a + 7\right)\cdot 13^{5} + \left(9 a^{3} + 3 a^{2} + 12 a\right)\cdot 13^{6} + \left(10 a^{2} + 8 a + 2\right)\cdot 13^{7} + \left(11 a^{3} + 9 a^{2} + 12 a + 6\right)\cdot 13^{8} + \left(12 a^{3} + 3 a^{2} + 10 a + 12\right)\cdot 13^{9} +O(13^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 9 a^{3} + 10 a^{2} + 9 a + 1 + \left(3 a^{3} + 8 a^{2} + 12 a + 10\right)\cdot 13 + \left(10 a^{3} + 2 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(4 a^{3} + 3 a^{2} + 12 a + 7\right)\cdot 13^{3} + \left(6 a^{3} + 9 a^{2} + 2 a\right)\cdot 13^{4} + \left(10 a^{3} + 4 a^{2} + 4 a + 2\right)\cdot 13^{5} + \left(12 a^{3} + 5 a^{2} + 8 a + 9\right)\cdot 13^{6} + \left(4 a^{3} + 10 a^{2} + 4 a + 7\right)\cdot 13^{7} + \left(4 a^{3} + 10 a^{2} + 12\right)\cdot 13^{8} + \left(10 a^{3} + 8 a^{2} + 6 a + 3\right)\cdot 13^{9} +O(13^{10})\)
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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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $16$ |
| $9$ | $2$ | $(1,5)(2,9)(3,8)(4,7)$ | $0$ |
| $36$ | $2$ | $(3,4)(5,8)(6,9)$ | $0$ |
| $8$ | $3$ | $(1,8,9)(2,3,5)(4,7,6)$ | $-2$ |
| $24$ | $3$ | $(1,5,6)(3,9,7)$ | $-2$ |
| $48$ | $3$ | $(1,2,3)(4,7,5)(6,8,9)$ | $1$ |
| $54$ | $4$ | $(1,4,5,7)(2,3,9,8)$ | $0$ |
| $72$ | $6$ | $(1,4,3)(2,5,6,7,9,8)$ | $0$ |
| $72$ | $6$ | $(1,3,2,8,7,6)(4,5)$ | $0$ |
| $54$ | $8$ | $(1,2,4,3,5,9,7,8)$ | $0$ |
| $54$ | $8$ | $(1,9,4,8,5,2,7,3)$ | $0$ |