Properties

Label 9.319873167719.16t1294.a
Dimension $9$
Group $S_4\wr C_2$
Conductor $319873167719$
Indicator $1$

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Basic invariants

Dimension:$9$
Group:$S_4\wr C_2$
Conductor:\(319873167719\)\(\medspace = 7^{3} \cdot 977^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.2.6377879282887.1
Galois orbit size: $1$
Smallest permutation container: 16T1294
Parity: odd
Projective image: $S_4\wr C_2$
Projective field: Galois closure of 8.2.6377879282887.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{3} + 2x + 9 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 2 + 4\cdot 11 + 8\cdot 11^{3} + 11^{4} + 10\cdot 11^{5} + 3\cdot 11^{6} + 2\cdot 11^{8} + 10\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 9 a^{2} + 9 a + 8 + \left(7 a^{2} + 10 a + 1\right)\cdot 11 + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{2} + \left(5 a^{2} + 9 a + 8\right)\cdot 11^{3} + \left(8 a^{2} + 9 a + 10\right)\cdot 11^{4} + \left(2 a^{2} + 3 a + 3\right)\cdot 11^{5} + \left(5 a^{2} + 2 a + 9\right)\cdot 11^{6} + \left(10 a^{2} + 5 a + 2\right)\cdot 11^{7} + \left(6 a^{2} + 2 a + 1\right)\cdot 11^{8} + \left(5 a^{2} + 7 a + 4\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a^{2} + 4 a + 6 + \left(2 a^{2} + 6 a + 1\right)\cdot 11 + \left(10 a^{2} + 6 a + 6\right)\cdot 11^{2} + \left(7 a^{2} + 5 a\right)\cdot 11^{3} + \left(2 a^{2} + 3\right)\cdot 11^{4} + \left(3 a + 4\right)\cdot 11^{5} + \left(9 a^{2} + 9 a + 3\right)\cdot 11^{6} + \left(10 a^{2} + 5 a + 3\right)\cdot 11^{7} + \left(10 a + 4\right)\cdot 11^{8} + \left(3 a^{2} + 10 a + 4\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 + 3\cdot 11 + 7\cdot 11^{2} + 4\cdot 11^{3} + 11^{4} + 2\cdot 11^{5} + 10\cdot 11^{6} + 8\cdot 11^{7} + 6\cdot 11^{8} +O(11^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a^{2} + 3 a + 2 + \left(8 a + 10\right)\cdot 11 + \left(a^{2} + 2 a + 9\right)\cdot 11^{2} + \left(6 a^{2} + 3 a + 2\right)\cdot 11^{3} + \left(5 a^{2} + 3\right)\cdot 11^{4} + \left(10 a^{2} + 3 a + 2\right)\cdot 11^{5} + \left(5 a^{2} + 9 a + 8\right)\cdot 11^{6} + \left(5 a^{2} + 5 a\right)\cdot 11^{7} + \left(a^{2} + 9 a + 7\right)\cdot 11^{8} + \left(4 a^{2} + 8 a + 1\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 9 a + 7 + \left(a^{2} + 4 a + 3\right)\cdot 11 + \left(2 a^{2} + a + 6\right)\cdot 11^{2} + \left(8 a^{2} + 7 a + 4\right)\cdot 11^{3} + \left(10 a^{2} + 6\right)\cdot 11^{4} + \left(7 a^{2} + 4 a + 3\right)\cdot 11^{5} + \left(7 a^{2} + 10 a + 5\right)\cdot 11^{6} + \left(10 a + 4\right)\cdot 11^{7} + \left(3 a^{2} + 8 a + 3\right)\cdot 11^{8} + \left(2 a^{2} + 3 a + 3\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 3 a + 9 + \left(8 a^{2} + 7 a + 5\right)\cdot 11 + \left(9 a^{2} + 10 a + 10\right)\cdot 11^{2} + \left(9 a^{2} + 7 a + 7\right)\cdot 11^{3} + \left(a^{2} + 6 a + 5\right)\cdot 11^{4} + \left(6 a^{2} + 3 a + 7\right)\cdot 11^{5} + \left(4 a^{2} + 5 a + 2\right)\cdot 11^{6} + \left(8 a^{2} + 7 a + 8\right)\cdot 11^{7} + \left(2 a^{2} + 7 a + 8\right)\cdot 11^{8} + \left(7 a^{2} + 8 a + 5\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 8 a^{2} + 5 a + 5 + \left(2 a^{2} + 6 a + 2\right)\cdot 11 + \left(8 a + 5\right)\cdot 11^{2} + \left(6 a^{2} + 10 a + 6\right)\cdot 11^{3} + \left(3 a^{2} + 3 a\right)\cdot 11^{4} + \left(5 a^{2} + 4 a + 10\right)\cdot 11^{5} + 7 a\cdot 11^{6} + \left(8 a^{2} + 8 a + 4\right)\cdot 11^{7} + \left(6 a^{2} + 4 a + 10\right)\cdot 11^{8} + \left(10 a^{2} + 4 a + 2\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 8 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character values
$c1$
$1$ $1$ $()$ $9$
$6$ $2$ $(1,3)(2,6)$ $-3$
$9$ $2$ $(1,3)(2,6)(4,7)(5,8)$ $1$
$12$ $2$ $(4,5)$ $3$
$24$ $2$ $(1,4)(2,5)(3,7)(6,8)$ $3$
$36$ $2$ $(1,2)(4,5)$ $1$
$36$ $2$ $(1,3)(2,6)(4,5)$ $-1$
$16$ $3$ $(4,7,8)$ $0$
$64$ $3$ $(2,3,6)(4,7,8)$ $0$
$12$ $4$ $(1,2,3,6)$ $-3$
$36$ $4$ $(1,2,3,6)(4,5,7,8)$ $1$
$36$ $4$ $(1,3)(2,6)(4,5,7,8)$ $1$
$72$ $4$ $(1,7,3,4)(2,8,6,5)$ $-1$
$72$ $4$ $(1,2,3,6)(4,5)$ $-1$
$144$ $4$ $(1,4,2,5)(3,7)(6,8)$ $1$
$48$ $6$ $(1,3)(2,6)(4,8,7)$ $0$
$96$ $6$ $(2,6,3)(4,5)$ $0$
$192$ $6$ $(1,5)(2,7,3,8,6,4)$ $0$
$144$ $8$ $(1,5,2,7,3,8,6,4)$ $-1$
$96$ $12$ $(1,2,3,6)(4,7,8)$ $0$
The blue line marks the conjugacy class containing complex conjugation.