Basic invariants
Dimension: | $8$ |
Group: | $S_3\wr S_3$ |
Conductor: | \(11851370496\)\(\medspace = 2^{12} \cdot 3^{10} \cdot 7^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin number field: | Galois closure of 9.1.186659085312.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 12T213 |
Parity: | even |
Projective image: | $S_3\wr S_3$ |
Projective field: | Galois closure of 9.1.186659085312.1 |
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 $\mathbb{Q}_{ 13 }$:
\( x^{4} + 3x^{2} + 12x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 10 a^{3} + 9 a^{2} + 12 a + 2 a\cdot 13 + \left(10 a^{3} + 12 a^{2} + 4 a + 5\right)\cdot 13^{2} + \left(3 a^{3} + 8 a^{2} + 6\right)\cdot 13^{3} + \left(7 a^{3} + 10 a^{2} + 3 a + 8\right)\cdot 13^{4} + \left(3 a^{3} + 12 a^{2} + 8\right)\cdot 13^{5} + \left(8 a^{3} + 7 a^{2} + 9 a + 1\right)\cdot 13^{6} + \left(9 a^{3} + 9 a^{2} + 11 a + 9\right)\cdot 13^{7} + \left(8 a^{3} + 6 a^{2} + 10\right)\cdot 13^{8} + \left(a^{3} + a^{2} + 6 a + 4\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 2 }$ | $=$ | \( 2 a^{3} + 11 a^{2} + 11 a + 12 + \left(8 a^{3} + 3 a^{2} + 5 a + 8\right)\cdot 13 + \left(10 a^{3} + 9 a + 6\right)\cdot 13^{2} + \left(a^{3} + 9 a^{2} + 2 a + 2\right)\cdot 13^{3} + \left(9 a^{3} + 2 a^{2} + 10 a\right)\cdot 13^{4} + \left(9 a^{2} + a + 12\right)\cdot 13^{5} + \left(10 a^{3} + 9\right)\cdot 13^{6} + \left(3 a^{3} + 6 a^{2} + 7 a + 5\right)\cdot 13^{7} + \left(12 a^{3} + 12 a^{2} + a + 4\right)\cdot 13^{8} + \left(11 a^{3} + 6 a^{2} + 10 a + 12\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 3 }$ | $=$ | \( 3 a^{3} + 4 a^{2} + a + 1 + \left(12 a^{3} + 12 a^{2} + 10 a + 10\right)\cdot 13 + \left(2 a^{3} + 8 a + 8\right)\cdot 13^{2} + \left(9 a^{3} + 4 a^{2} + 12 a + 2\right)\cdot 13^{3} + \left(5 a^{3} + 2 a^{2} + 9 a + 1\right)\cdot 13^{4} + \left(9 a^{3} + 12 a + 10\right)\cdot 13^{5} + \left(4 a^{3} + 5 a^{2} + 3 a + 10\right)\cdot 13^{6} + \left(3 a^{3} + 3 a^{2} + a + 1\right)\cdot 13^{7} + \left(4 a^{3} + 6 a^{2} + 12 a + 2\right)\cdot 13^{8} + \left(11 a^{3} + 11 a^{2} + 6 a + 9\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 4 }$ | $=$ | \( 3 + 6\cdot 13^{2} + 3\cdot 13^{3} + 8\cdot 13^{4} + 9\cdot 13^{5} + 13^{8} +O(13^{10})\) |
$r_{ 5 }$ | $=$ | \( 4 a^{3} + a^{2} + 10 a + 4 + \left(12 a^{3} + 11 a^{2} + 7 a + 9\right)\cdot 13 + \left(12 a^{3} + 12 a^{2} + 9 a + 5\right)\cdot 13^{2} + \left(6 a^{3} + 8 a^{2} + 7 a + 1\right)\cdot 13^{3} + \left(4 a^{3} + 12 a^{2} + 8 a + 2\right)\cdot 13^{4} + \left(8 a^{3} + 2 a^{2} + 12 a + 10\right)\cdot 13^{5} + \left(5 a^{3} + a^{2} + 7 a + 7\right)\cdot 13^{6} + \left(9 a^{3} + 8 a^{2} + 7 a + 3\right)\cdot 13^{7} + \left(4 a^{3} + 10 a^{2} + 6 a + 2\right)\cdot 13^{8} + \left(11 a^{3} + 7 a^{2} + 5 a + 6\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 6 }$ | $=$ | \( 8 a^{3} + 11 a^{2} + 9 a + 1 + \left(12 a^{3} + 8 a^{2} + 10 a + 11\right)\cdot 13 + \left(a^{3} + 8 a^{2} + 8 a + 12\right)\cdot 13^{2} + \left(11 a^{3} + 12 a + 8\right)\cdot 13^{3} + \left(10 a^{3} + 12 a^{2} + 4 a + 3\right)\cdot 13^{4} + \left(a^{3} + 12 a^{2} + 12 a + 8\right)\cdot 13^{5} + \left(9 a^{3} + 3 a^{2} + 10 a + 6\right)\cdot 13^{6} + \left(11 a^{3} + 10 a^{2} + 5 a + 5\right)\cdot 13^{7} + \left(10 a^{3} + a^{2} + 2 a + 8\right)\cdot 13^{8} + \left(9 a^{3} + 10 a^{2} + 12 a + 4\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 7 }$ | $=$ | \( 9 a^{3} + 12 a^{2} + 3 a + 7 + \left(a^{2} + 5 a + 1\right)\cdot 13 + \left(3 a + 7\right)\cdot 13^{2} + \left(6 a^{3} + 4 a^{2} + 5 a + 4\right)\cdot 13^{3} + \left(8 a^{3} + 4 a + 12\right)\cdot 13^{4} + \left(4 a^{3} + 10 a^{2} + 6\right)\cdot 13^{5} + \left(7 a^{3} + 11 a^{2} + 5 a + 6\right)\cdot 13^{6} + \left(3 a^{3} + 4 a^{2} + 5 a + 4\right)\cdot 13^{7} + \left(8 a^{3} + 2 a^{2} + 6 a + 2\right)\cdot 13^{8} + \left(a^{3} + 5 a^{2} + 7 a + 12\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 8 }$ | $=$ | \( 5 a^{3} + 7 a^{2} + 7 + \left(8 a^{3} + 4 a^{2} + 5 a + 5\right)\cdot 13 + \left(4 a^{3} + 10 a^{2} + a\right)\cdot 13^{2} + \left(5 a^{3} + 7 a^{2}\right)\cdot 13^{3} + \left(3 a^{3} + 7 a + 4\right)\cdot 13^{4} + \left(10 a^{3} + 6 a^{2} + 12 a + 2\right)\cdot 13^{5} + \left(5 a^{3} + 10 a^{2} + 10 a + 12\right)\cdot 13^{6} + \left(12 a^{3} + 2 a^{2} + 11 a\right)\cdot 13^{7} + \left(6 a^{3} + 9 a^{2} + 10\right)\cdot 13^{8} + \left(6 a^{3} + 6 a^{2} + 3 a + 8\right)\cdot 13^{9} +O(13^{10})\) |
$r_{ 9 }$ | $=$ | \( 11 a^{3} + 10 a^{2} + 6 a + 7 + \left(9 a^{3} + 8 a^{2} + 4 a + 5\right)\cdot 13 + \left(8 a^{3} + 6 a^{2} + 6 a + 12\right)\cdot 13^{2} + \left(7 a^{3} + 8 a^{2} + 10 a + 8\right)\cdot 13^{3} + \left(2 a^{3} + 10 a^{2} + 3 a + 11\right)\cdot 13^{4} + \left(10 a^{2} + 12 a + 9\right)\cdot 13^{5} + \left(a^{3} + 10 a^{2} + 3 a + 8\right)\cdot 13^{6} + \left(11 a^{3} + 6 a^{2} + a + 7\right)\cdot 13^{7} + \left(8 a^{3} + 2 a^{2} + 8 a + 10\right)\cdot 13^{8} + \left(10 a^{3} + 2 a^{2} + 6\right)\cdot 13^{9} +O(13^{10})\) |
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 |
$c1$ | |||
$1$ | $1$ | $()$ | $8$ |
$9$ | $2$ | $(5,6)$ | $0$ |
$18$ | $2$ | $(1,5)(3,6)(4,9)$ | $4$ |
$27$ | $2$ | $(1,3)(2,7)(5,6)$ | $0$ |
$27$ | $2$ | $(1,3)(5,6)$ | $0$ |
$54$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
$6$ | $3$ | $(2,7,8)$ | $-4$ |
$8$ | $3$ | $(1,3,4)(2,7,8)(5,6,9)$ | $-1$ |
$12$ | $3$ | $(1,3,4)(2,7,8)$ | $2$ |
$72$ | $3$ | $(1,2,5)(3,7,6)(4,8,9)$ | $2$ |
$54$ | $4$ | $(1,5,3,6)(4,9)$ | $0$ |
$162$ | $4$ | $(2,5,7,6)(3,4)(8,9)$ | $0$ |
$36$ | $6$ | $(1,5)(2,7,8)(3,6)(4,9)$ | $-2$ |
$36$ | $6$ | $(2,6,7,9,8,5)$ | $-2$ |
$36$ | $6$ | $(2,7,8)(5,6)$ | $0$ |
$36$ | $6$ | $(1,3,4)(2,7,8)(5,6)$ | $0$ |
$54$ | $6$ | $(1,3)(2,8,7)(5,6)$ | $0$ |
$72$ | $6$ | $(1,6,3,9,4,5)(2,7,8)$ | $1$ |
$108$ | $6$ | $(1,2,3,7,4,8)(5,6)$ | $0$ |
$216$ | $6$ | $(1,2,5,3,7,6)(4,8,9)$ | $0$ |
$144$ | $9$ | $(1,2,6,3,7,9,4,8,5)$ | $-1$ |
$108$ | $12$ | $(1,5,3,6)(2,7,8)(4,9)$ | $0$ |