Properties

Label 8.15551087616.12t213.a
Dimension $8$
Group $S_3\wr S_3$
Conductor $15551087616$
Indicator $1$

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Basic invariants

Dimension:$8$
Group:$S_3\wr S_3$
Conductor:\(15551087616\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 433^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 9.1.561135078144.1
Galois orbit size: $1$
Smallest permutation container: 12T213
Parity: even
Projective image: $S_3\wr S_3$
Projective field: Galois closure of 9.1.561135078144.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{3} + x + 40 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 29 a^{2} + 10 a + 20 + \left(35 a^{2} + 24 a + 27\right)\cdot 43 + \left(5 a^{2} + 30 a + 13\right)\cdot 43^{2} + \left(34 a + 27\right)\cdot 43^{3} + \left(3 a^{2} + 4\right)\cdot 43^{4} + \left(15 a^{2} + 7 a + 23\right)\cdot 43^{5} + \left(27 a^{2} + 35 a + 2\right)\cdot 43^{6} + \left(21 a^{2} + 24 a + 7\right)\cdot 43^{7} + \left(29 a^{2} + 34 a + 29\right)\cdot 43^{8} + \left(4 a^{2} + 15 a + 26\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a^{2} + 14 a + 1 + \left(6 a^{2} + 3 a + 1\right)\cdot 43 + \left(10 a^{2} + 27 a + 36\right)\cdot 43^{2} + \left(23 a^{2} + 3 a + 39\right)\cdot 43^{3} + \left(5 a^{2} + 15 a + 26\right)\cdot 43^{4} + \left(36 a^{2} + 38 a + 15\right)\cdot 43^{5} + \left(15 a^{2} + 5 a + 15\right)\cdot 43^{6} + \left(20 a^{2} + 36 a + 30\right)\cdot 43^{7} + \left(22 a^{2} + 33 a + 37\right)\cdot 43^{8} + \left(36 a^{2} + 4 a + 3\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a^{2} + a + \left(20 a^{2} + 29 a + 13\right)\cdot 43 + \left(22 a^{2} + 19 a + 33\right)\cdot 43^{2} + \left(3 a^{2} + 42 a + 36\right)\cdot 43^{3} + \left(37 a^{2} + 41\right)\cdot 43^{4} + \left(14 a^{2} + 32 a + 33\right)\cdot 43^{5} + \left(32 a^{2} + 29 a + 3\right)\cdot 43^{6} + \left(10 a^{2} + 40 a + 12\right)\cdot 43^{7} + \left(11 a^{2} + 14 a + 18\right)\cdot 43^{8} + \left(28 a^{2} + 36 a + 1\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 38 a^{2} + 23 a + 5 + \left(16 a^{2} + 3 a + 25\right)\cdot 43 + \left(12 a + 18\right)\cdot 43^{2} + \left(29 a^{2} + 8 a + 39\right)\cdot 43^{3} + \left(28 a^{2} + 17 a + 21\right)\cdot 43^{4} + \left(42 a^{2} + 4 a + 9\right)\cdot 43^{5} + \left(35 a^{2} + 40 a + 6\right)\cdot 43^{6} + \left(a^{2} + 20 a + 6\right)\cdot 43^{7} + \left(3 a^{2} + 30 a + 27\right)\cdot 43^{8} + \left(16 a^{2} + 30 a + 7\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 37 a^{2} + 14 a + 30 + \left(16 a^{2} + 34 a + 36\right)\cdot 43 + \left(38 a^{2} + 19 a + 11\right)\cdot 43^{2} + \left(10 a^{2} + 8 a + 17\right)\cdot 43^{3} + \left(24 a^{2} + 42 a + 39\right)\cdot 43^{4} + \left(29 a^{2} + 16 a + 39\right)\cdot 43^{5} + \left(36 a^{2} + 38 a + 14\right)\cdot 43^{6} + \left(39 a^{2} + 25 a\right)\cdot 43^{7} + \left(30 a^{2} + 21 a + 29\right)\cdot 43^{8} + \left(12 a^{2} + 16\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 39 a^{2} + 19 a + 20 + \left(5 a^{2} + 10 a + 3\right)\cdot 43 + \left(20 a^{2} + 11 a + 3\right)\cdot 43^{2} + \left(10 a^{2} + 35 a + 27\right)\cdot 43^{3} + \left(20 a^{2} + 24 a + 30\right)\cdot 43^{4} + \left(28 a^{2} + 6 a + 28\right)\cdot 43^{5} + \left(17 a^{2} + 16 a + 22\right)\cdot 43^{6} + \left(30 a^{2} + 24 a + 39\right)\cdot 43^{7} + \left(28 a^{2} + 40 a + 29\right)\cdot 43^{8} + \left(41 a^{2} + 18 a + 24\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 15 a^{2} + 22 a + 25 + \left(21 a^{2} + 33 a + 3\right)\cdot 43 + \left(24 a^{2} + 31 a + 26\right)\cdot 43^{2} + \left(35 a^{2} + 11 a + 36\right)\cdot 43^{3} + \left(33 a^{2} + 22 a + 10\right)\cdot 43^{4} + \left(35 a^{2} + 39 a + 8\right)\cdot 43^{5} + \left(21 a^{2} + 36 a + 13\right)\cdot 43^{6} + \left(9 a^{2} + 4 a + 13\right)\cdot 43^{7} + \left(32 a^{2} + 15 a + 2\right)\cdot 43^{8} + \left(35 a + 24\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 42 a^{2} + 11 a + \left(28 a^{2} + 28 a + 23\right)\cdot 43 + \left(12 a^{2} + 23 a + 32\right)\cdot 43^{2} + \left(7 a^{2} + 39 a + 17\right)\cdot 43^{3} + \left(6 a^{2} + 19 a + 35\right)\cdot 43^{4} + \left(35 a^{2} + 39 a + 7\right)\cdot 43^{5} + \left(36 a^{2} + 13 a + 23\right)\cdot 43^{6} + \left(11 a^{2} + 13 a\right)\cdot 43^{7} + \left(24 a^{2} + 36 a + 40\right)\cdot 43^{8} + \left(37 a^{2} + 34 a + 19\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 34 a^{2} + 15 a + 28 + \left(19 a^{2} + 5 a + 38\right)\cdot 43 + \left(37 a^{2} + 39 a + 39\right)\cdot 43^{2} + \left(8 a^{2} + 30 a + 15\right)\cdot 43^{3} + \left(13 a^{2} + 28 a + 3\right)\cdot 43^{4} + \left(20 a^{2} + 30 a + 5\right)\cdot 43^{5} + \left(33 a^{2} + 41 a + 27\right)\cdot 43^{6} + \left(25 a^{2} + 23 a + 19\right)\cdot 43^{7} + \left(32 a^{2} + 30 a + 1\right)\cdot 43^{8} + \left(36 a^{2} + 37 a + 4\right)\cdot 43^{9} +O(43^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

Cycle notation
$(1,2,4)(3,5,6)(7,8,9)$
$(2,4)(3,5)(7,8)$
$(4,5,8)$
$(1,6,9)$
$(2,3)$
$(2,3,7)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 9 }$ Character values
$c1$
$1$ $1$ $()$ $8$
$9$ $2$ $(2,3)$ $0$
$18$ $2$ $(2,4)(3,5)(7,8)$ $4$
$27$ $2$ $(1,6)(2,3)(4,5)$ $0$
$27$ $2$ $(1,6)(2,3)$ $0$
$54$ $2$ $(1,2)(3,6)(4,5)(7,9)$ $0$
$6$ $3$ $(1,6,9)$ $-4$
$8$ $3$ $(1,9,6)(2,7,3)(4,8,5)$ $-1$
$12$ $3$ $(1,9,6)(2,7,3)$ $2$
$72$ $3$ $(1,2,4)(3,5,6)(7,8,9)$ $2$
$54$ $4$ $(1,3,6,2)(7,9)$ $0$
$162$ $4$ $(1,3,6,2)(4,5)(7,9)$ $0$
$36$ $6$ $(1,6,9)(2,4)(3,5)(7,8)$ $-2$
$36$ $6$ $(1,7,9,3,6,2)$ $-2$
$36$ $6$ $(1,6,9)(2,3)$ $0$
$36$ $6$ $(1,6,9)(2,3)(4,5,8)$ $0$
$54$ $6$ $(1,9,6)(2,3)(4,5)$ $0$
$72$ $6$ $(1,6,9)(2,4,7,8,3,5)$ $1$
$108$ $6$ $(1,7,9,3,6,2)(4,5)$ $0$
$216$ $6$ $(1,3,5,6,2,4)(7,8,9)$ $0$
$144$ $9$ $(1,7,8,9,3,5,6,2,4)$ $-1$
$108$ $12$ $(1,6,9)(2,4,3,5)(7,8)$ $0$
The blue line marks the conjugacy class containing complex conjugation.