Properties

Label 8.6591978481.12t213.a
Dimension $8$
Group $S_3\wr S_3$
Conductor $6591978481$
Indicator $1$

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Basic invariants

Dimension:$8$
Group:$S_3\wr S_3$
Conductor:\(6591978481\)\(\medspace = 11^{6} \cdot 61^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 9.1.3323228821.1
Galois orbit size: $1$
Smallest permutation container: 12T213
Parity: even
Projective image: $S_3\wr S_3$
Projective field: Galois closure of 9.1.3323228821.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{4} + 3x^{2} + 19x + 5 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 5 a^{3} + 10 a^{2} + 10 a + 2 + \left(15 a^{3} + 20 a^{2} + 6 a + 21\right)\cdot 23 + \left(12 a^{3} + 16 a^{2} + 7 a + 2\right)\cdot 23^{2} + \left(16 a^{3} + 21 a^{2} + 9 a + 4\right)\cdot 23^{3} + \left(7 a^{3} + 21 a^{2} + 7 a + 16\right)\cdot 23^{4} + \left(22 a^{3} + 13 a^{2} + 7 a + 19\right)\cdot 23^{5} + \left(a^{3} + 20 a^{2} + 4 a + 6\right)\cdot 23^{6} + \left(16 a^{3} + 10 a^{2} + 3\right)\cdot 23^{7} + \left(9 a^{3} + 8 a^{2} + 11 a + 14\right)\cdot 23^{8} + \left(15 a^{3} + 20 a^{2} + 12 a + 20\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 16 a^{3} + 11 a^{2} + 6 a + 13 + \left(16 a^{3} + 9 a^{2} + 10 a + 6\right)\cdot 23 + \left(22 a^{3} + 13 a^{2} + 2 a + 11\right)\cdot 23^{2} + \left(2 a^{3} + 16 a^{2} + 18 a + 15\right)\cdot 23^{3} + \left(14 a^{3} + 11 a + 10\right)\cdot 23^{4} + \left(7 a^{3} + 21 a^{2} + 19 a + 4\right)\cdot 23^{5} + \left(10 a^{3} + 20 a^{2} + 15 a + 13\right)\cdot 23^{6} + \left(a^{3} + 20 a^{2} + 16 a + 2\right)\cdot 23^{7} + \left(7 a^{3} + 9 a^{2} + 11 a + 1\right)\cdot 23^{8} + \left(13 a^{3} + 8 a^{2} + 21 a + 3\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a^{3} + 4 a^{2} + 4 a + 4 + \left(11 a^{3} + 18 a^{2} + 17 a + 15\right)\cdot 23 + \left(5 a^{3} + 6 a^{2} + 2 a + 17\right)\cdot 23^{2} + \left(22 a^{3} + a^{2} + 17 a + 8\right)\cdot 23^{3} + \left(4 a^{3} + 3 a + 17\right)\cdot 23^{4} + \left(6 a^{3} + 9 a^{2} + 7 a + 16\right)\cdot 23^{5} + \left(16 a^{3} + 8 a^{2} + 15 a + 11\right)\cdot 23^{6} + \left(17 a^{3} + 4 a^{2} + 6 a + 18\right)\cdot 23^{7} + \left(18 a^{3} + 3 a^{2} + 13 a\right)\cdot 23^{8} + \left(8 a^{3} + 12 a^{2} + 2 a + 19\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 20 + 15\cdot 23 + 14\cdot 23^{2} + 22\cdot 23^{3} + 11\cdot 23^{4} + 6\cdot 23^{5} + 12\cdot 23^{6} + 23^{7} + 9\cdot 23^{8} + 22\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 18 a^{3} + 13 a^{2} + 13 a + 2 + \left(7 a^{3} + 2 a^{2} + 16 a + 9\right)\cdot 23 + \left(10 a^{3} + 6 a^{2} + 15 a + 5\right)\cdot 23^{2} + \left(6 a^{3} + a^{2} + 13 a + 19\right)\cdot 23^{3} + \left(15 a^{3} + a^{2} + 15 a + 17\right)\cdot 23^{4} + \left(9 a^{2} + 15 a + 19\right)\cdot 23^{5} + \left(21 a^{3} + 2 a^{2} + 18 a + 3\right)\cdot 23^{6} + \left(6 a^{3} + 12 a^{2} + 22 a + 18\right)\cdot 23^{7} + \left(13 a^{3} + 14 a^{2} + 11 a + 22\right)\cdot 23^{8} + \left(7 a^{3} + 2 a^{2} + 10 a + 2\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 9 a^{3} + 16 a^{2} + 21 a + 7 + \left(17 a^{3} + 8 a^{2} + 6 a + 15\right)\cdot 23 + \left(5 a^{3} + 16 a^{2} + 9\right)\cdot 23^{2} + \left(19 a^{3} + 7 a^{2} + 22 a + 15\right)\cdot 23^{3} + \left(13 a^{3} + 22 a^{2} + 14 a + 21\right)\cdot 23^{4} + \left(21 a^{3} + 10 a^{2} + 10 a + 21\right)\cdot 23^{5} + \left(5 a^{3} + 10 a^{2} + 22 a + 3\right)\cdot 23^{6} + \left(16 a^{3} + 6 a^{2} + 12 a + 2\right)\cdot 23^{7} + \left(11 a^{3} + 16 a^{2} + a + 2\right)\cdot 23^{8} + \left(18 a^{3} + 3 a^{2} + 4 a + 1\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 4 a^{3} + 19 a^{2} + 10 a + 15 + \left(6 a^{3} + 17 a^{2} + 5 a + 18\right)\cdot 23 + \left(14 a^{3} + a^{2} + 16 a + 22\right)\cdot 23^{2} + \left(22 a^{3} + 19 a^{2} + 15 a + 22\right)\cdot 23^{3} + \left(16 a^{3} + 15 a^{2} + 9 a + 4\right)\cdot 23^{4} + \left(13 a^{3} + 8 a^{2} + 17 a + 4\right)\cdot 23^{5} + \left(a^{3} + 14 a^{2} + 9 a + 11\right)\cdot 23^{6} + \left(7 a^{3} + 21 a^{2} + 17 a + 20\right)\cdot 23^{7} + \left(6 a^{3} + 13 a^{2} + 21 a + 7\right)\cdot 23^{8} + \left(22 a^{3} + 17 a^{2} + 8 a + 12\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 17 a^{3} + 9 a + 5 + \left(5 a^{3} + 10 a^{2} + 12\right)\cdot 23 + \left(3 a^{3} + 14 a^{2} + 4 a + 5\right)\cdot 23^{2} + \left(a^{3} + 2 a^{2} + 13 a + 14\right)\cdot 23^{3} + \left(a^{3} + 7 a^{2} + 9 a\right)\cdot 23^{4} + \left(3 a^{3} + 5 a^{2} + 21 a + 2\right)\cdot 23^{5} + \left(5 a^{3} + 20 a\right)\cdot 23^{6} + \left(21 a^{3} + 20 a^{2} + 21 a + 7\right)\cdot 23^{7} + \left(20 a^{3} + 5 a^{2} + 10 a + 14\right)\cdot 23^{8} + \left(14 a^{3} + 16 a^{2} + 11 a + 14\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 21 a^{3} + 19 a^{2} + 19 a + 4 + \left(11 a^{3} + 4 a^{2} + 5 a + 1\right)\cdot 23 + \left(17 a^{3} + 16 a^{2} + 20 a + 2\right)\cdot 23^{2} + \left(21 a^{2} + 5 a + 15\right)\cdot 23^{3} + \left(18 a^{3} + 22 a^{2} + 19 a + 13\right)\cdot 23^{4} + \left(16 a^{3} + 13 a^{2} + 15 a + 19\right)\cdot 23^{5} + \left(6 a^{3} + 14 a^{2} + 7 a + 5\right)\cdot 23^{6} + \left(5 a^{3} + 18 a^{2} + 16 a + 18\right)\cdot 23^{7} + \left(4 a^{3} + 19 a^{2} + 9 a + 19\right)\cdot 23^{8} + \left(14 a^{3} + 10 a^{2} + 20 a + 18\right)\cdot 23^{9} +O(23^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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