Basic invariants
| Dimension: | $16$ |
| Group: | $((C_3^2:Q_8):C_3):C_2$ |
| Conductor: | \(525\!\cdots\!625\)\(\medspace = 3^{8} \cdot 5^{10} \cdot 31^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.3.374415615421875.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 24T1334 |
| Parity: | even |
| Projective image: | $C_3^2:\GL(2,3)$ |
| Projective field: | Galois closure of 9.3.374415615421875.1 |
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:
| $r_{ 1 }$ | $=$ |
\( 8 + 4\cdot 13 + 11\cdot 13^{3} + 2\cdot 13^{4} + 12\cdot 13^{5} + 3\cdot 13^{6} + 5\cdot 13^{7} +O(13^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 9 a^{3} + 8 a^{2} + 6 + \left(8 a^{3} + 2 a^{2} + 7 a + 2\right)\cdot 13 + \left(10 a^{3} + 4 a^{2} + 7\right)\cdot 13^{2} + \left(8 a^{3} + 8 a^{2} + 6 a + 11\right)\cdot 13^{3} + \left(7 a^{3} + 12 a^{2} + 10 a + 3\right)\cdot 13^{4} + \left(9 a^{3} + 7 a^{2} + 8 a + 10\right)\cdot 13^{5} + \left(7 a^{3} + 3 a^{2}\right)\cdot 13^{6} + \left(2 a^{3} + 9 a^{2} + 9 a + 10\right)\cdot 13^{7} + \left(a^{3} + 6 a^{2} + 4 a + 9\right)\cdot 13^{8} + \left(11 a^{3} + 11 a + 6\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 5 a^{3} + 2 a^{2} + 9 a + \left(10 a^{3} + 4 a^{2} + 5 a + 7\right)\cdot 13 + \left(9 a^{3} + 9 a^{2} + 11 a\right)\cdot 13^{2} + \left(5 a^{3} + 5 a^{2} + 3 a + 6\right)\cdot 13^{3} + \left(2 a^{3} + 5 a + 3\right)\cdot 13^{4} + \left(10 a^{3} + 3 a^{2} + 3 a + 8\right)\cdot 13^{5} + \left(11 a^{3} + 11 a^{2} + 2 a + 9\right)\cdot 13^{6} + \left(6 a^{3} + 4 a^{2} + 9\right)\cdot 13^{7} + \left(a^{3} + 8 a^{2} + 11 a + 8\right)\cdot 13^{8} + \left(4 a^{3} + 10 a^{2} + 2 a + 4\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 5 a^{3} + 6 a + 4 + \left(11 a^{3} + a^{2} + 2 a + 10\right)\cdot 13 + \left(3 a^{3} + 10 a^{2} + 5 a + 11\right)\cdot 13^{2} + \left(7 a^{3} + 4 a^{2} + 6 a\right)\cdot 13^{3} + \left(2 a^{3} + 11 a^{2} + 12 a\right)\cdot 13^{4} + \left(a^{3} + 2 a^{2} + 2 a + 9\right)\cdot 13^{5} + \left(5 a^{3} + 7 a + 5\right)\cdot 13^{6} + \left(10 a^{3} + 12 a^{2} + 12 a + 7\right)\cdot 13^{7} + \left(5 a^{3} + 7 a^{2} + a + 3\right)\cdot 13^{8} + \left(9 a^{3} + a^{2} + 12\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 11 a^{3} + 5 a^{2} + 5 a + \left(5 a^{3} + 8 a^{2} + 8 a + 5\right)\cdot 13 + \left(8 a^{3} + a^{2} + 5 a + 9\right)\cdot 13^{2} + \left(4 a^{3} + 8 a^{2} + 12 a + 12\right)\cdot 13^{3} + \left(4 a^{3} + 3 a^{2} + 9 a + 5\right)\cdot 13^{4} + \left(12 a^{3} + 7 a + 10\right)\cdot 13^{5} + \left(5 a^{3} + 9 a^{2} + 3 a + 5\right)\cdot 13^{6} + \left(9 a^{3} + 12 a^{2} + 5 a + 5\right)\cdot 13^{7} + \left(a^{3} + 12 a^{2} + 8 a + 4\right)\cdot 13^{8} + \left(5 a^{3} + 6 a^{2} + 9 a + 8\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 6 a^{3} + 9 a^{2} + 11 a + 7 + \left(5 a + 2\right)\cdot 13 + \left(3 a^{3} + 8 a^{2} + 7 a + 1\right)\cdot 13^{2} + \left(6 a^{3} + 11 a^{2} + 7 a + 8\right)\cdot 13^{3} + \left(4 a^{2} + 7 a + 10\right)\cdot 13^{4} + \left(11 a^{3} + 3 a^{2}\right)\cdot 13^{5} + \left(3 a^{3} + 4 a^{2} + a + 7\right)\cdot 13^{6} + \left(7 a^{3} + 5 a^{2} + 4 a + 8\right)\cdot 13^{7} + \left(7 a^{2} + 3 a + 7\right)\cdot 13^{8} + \left(3 a^{3} + 4 a^{2} + 9 a + 4\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 7 }$ |
$=$ |
\( a^{3} + 9 a^{2} + 1 + \left(5 a^{3} + a^{2} + 10 a\right)\cdot 13 + \left(8 a^{3} + 12 a^{2} + a + 10\right)\cdot 13^{2} + \left(11 a^{3} + 8 a^{2} + 3 a\right)\cdot 13^{3} + \left(11 a^{3} + 5 a^{2} + 6 a + 4\right)\cdot 13^{4} + \left(4 a^{3} + 11 a^{2} + 8 a + 3\right)\cdot 13^{5} + \left(8 a^{3} + 12 a^{2} + 12 a + 2\right)\cdot 13^{6} + \left(9 a^{3} + 5 a^{2} + 11\right)\cdot 13^{7} + \left(12 a^{3} + 3 a^{2} + 2 a\right)\cdot 13^{8} + \left(9 a^{3} + 12 a^{2} + 3 a + 7\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 8 }$ |
$=$ |
\( a^{3} + 11 a^{2} + 12 a + 10 + \left(a^{3} + 10 a^{2} + 4 a + 4\right)\cdot 13 + \left(10 a^{3} + 10 a^{2} + 8 a + 5\right)\cdot 13^{2} + \left(6 a^{3} + 3 a^{2} + 3 a + 12\right)\cdot 13^{3} + \left(11 a^{3} + 9 a^{2}\right)\cdot 13^{4} + \left(6 a^{3} + a^{2} + 6 a + 9\right)\cdot 13^{5} + \left(2 a^{2} + 6 a + 11\right)\cdot 13^{6} + \left(7 a^{3} + 12 a^{2} + 11 a + 8\right)\cdot 13^{7} + \left(8 a^{3} + 10 a^{2} + a + 10\right)\cdot 13^{8} + \left(5 a^{3} + 7 a^{2} + 2 a + 7\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 9 }$ |
$=$ |
\( a^{3} + 8 a^{2} + 9 a + 6 + \left(9 a^{3} + 9 a^{2} + 7 a + 2\right)\cdot 13 + \left(10 a^{3} + 8 a^{2} + 11 a + 6\right)\cdot 13^{2} + \left(8 a + 1\right)\cdot 13^{3} + \left(11 a^{3} + 4 a^{2} + 12 a + 7\right)\cdot 13^{4} + \left(8 a^{3} + 8 a^{2} + 1\right)\cdot 13^{5} + \left(8 a^{3} + 8 a^{2} + 5 a + 5\right)\cdot 13^{6} + \left(11 a^{3} + 2 a^{2} + 8 a + 11\right)\cdot 13^{7} + \left(6 a^{3} + 7 a^{2} + 5 a + 5\right)\cdot 13^{8} + \left(3 a^{3} + 7 a^{2}\right)\cdot 13^{9} +O(13^{10})\)
| |
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 |
| $c1$ | |||
| $1$ | $1$ | $()$ | $16$ |
| $9$ | $2$ | $(1,6)(2,8)(3,9)(4,5)$ | $0$ |
| $36$ | $2$ | $(4,6)(5,8)(7,9)$ | $0$ |
| $8$ | $3$ | $(1,3,2)(4,5,7)(6,8,9)$ | $-2$ |
| $24$ | $3$ | $(1,8,5)(3,7,9)$ | $-2$ |
| $48$ | $3$ | $(1,9,2)(3,4,5)(6,8,7)$ | $1$ |
| $54$ | $4$ | $(1,2,6,8)(3,4,9,5)$ | $0$ |
| $72$ | $6$ | $(1,8,9,7,4,2)(3,5,6)$ | $0$ |
| $72$ | $6$ | $(2,8,9,3,5,4)(6,7)$ | $0$ |
| $54$ | $8$ | $(1,5,2,3,6,4,8,9)$ | $0$ |
| $54$ | $8$ | $(1,4,2,9,6,5,8,3)$ | $0$ |