Properties

Label 9.462786790375.16t1294.a
Dimension $9$
Group $S_4\wr C_2$
Conductor $462786790375$
Indicator $1$

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Basic invariants

Dimension:$9$
Group:$S_4\wr C_2$
Conductor:\(462786790375\)\(\medspace = 5^{3} \cdot 7^{3} \cdot 13^{3} \cdot 17^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 8.2.83490514835.1
Galois orbit size: $1$
Smallest permutation container: 16T1294
Parity: odd
Projective image: $S_4\wr C_2$
Projective field: Galois closure of 8.2.83490514835.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: \( x^{2} + 49x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 14 + 6\cdot 53 + 42\cdot 53^{2} + 38\cdot 53^{3} + 49\cdot 53^{4} + 14\cdot 53^{5} + 37\cdot 53^{6} + 14\cdot 53^{7} + 16\cdot 53^{8} + 45\cdot 53^{9} +O(53^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 15 a + 2 + \left(49 a + 29\right)\cdot 53 + \left(22 a + 29\right)\cdot 53^{2} + \left(22 a + 50\right)\cdot 53^{3} + \left(12 a + 43\right)\cdot 53^{4} + \left(27 a + 5\right)\cdot 53^{5} + \left(33 a + 49\right)\cdot 53^{6} + \left(51 a + 19\right)\cdot 53^{7} + \left(3 a + 2\right)\cdot 53^{8} + \left(7 a + 33\right)\cdot 53^{9} +O(53^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 33 a + 22 + \left(15 a + 6\right)\cdot 53 + \left(24 a + 48\right)\cdot 53^{2} + \left(a + 14\right)\cdot 53^{3} + \left(22 a + 26\right)\cdot 53^{4} + \left(35 a + 13\right)\cdot 53^{5} + \left(25 a + 41\right)\cdot 53^{6} + \left(46 a + 22\right)\cdot 53^{7} + \left(4 a + 27\right)\cdot 53^{8} + \left(9 a + 3\right)\cdot 53^{9} +O(53^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 31 a + 39 + \left(46 a + 40\right)\cdot 53 + \left(21 a + 34\right)\cdot 53^{2} + \left(11 a + 36\right)\cdot 53^{3} + \left(7 a + 39\right)\cdot 53^{4} + \left(50 a + 34\right)\cdot 53^{5} + \left(22 a + 35\right)\cdot 53^{6} + \left(37 a + 15\right)\cdot 53^{7} + \left(31 a + 50\right)\cdot 53^{8} + \left(12 a + 24\right)\cdot 53^{9} +O(53^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 22 a + 4 + \left(6 a + 37\right)\cdot 53 + \left(31 a + 22\right)\cdot 53^{2} + \left(41 a + 7\right)\cdot 53^{3} + \left(45 a + 4\right)\cdot 53^{4} + \left(2 a + 16\right)\cdot 53^{5} + \left(30 a + 24\right)\cdot 53^{6} + \left(15 a + 36\right)\cdot 53^{7} + \left(21 a + 33\right)\cdot 53^{8} + \left(40 a + 43\right)\cdot 53^{9} +O(53^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 20 a + 48 + \left(37 a + 35\right)\cdot 53 + \left(28 a + 23\right)\cdot 53^{2} + \left(51 a + 49\right)\cdot 53^{3} + \left(30 a + 6\right)\cdot 53^{4} + \left(17 a + 27\right)\cdot 53^{5} + \left(27 a + 2\right)\cdot 53^{6} + \left(6 a + 24\right)\cdot 53^{7} + 48 a\cdot 53^{8} + \left(43 a + 35\right)\cdot 53^{9} +O(53^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 23 + 4\cdot 53 + 45\cdot 53^{2} + 2\cdot 53^{3} + 23\cdot 53^{4} + 50\cdot 53^{5} + 24\cdot 53^{6} + 44\cdot 53^{7} + 8\cdot 53^{8} + 22\cdot 53^{9} +O(53^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 38 a + 9 + \left(3 a + 52\right)\cdot 53 + \left(30 a + 18\right)\cdot 53^{2} + \left(30 a + 11\right)\cdot 53^{3} + \left(40 a + 18\right)\cdot 53^{4} + \left(25 a + 49\right)\cdot 53^{5} + \left(19 a + 49\right)\cdot 53^{6} + \left(a + 33\right)\cdot 53^{7} + \left(49 a + 19\right)\cdot 53^{8} + \left(45 a + 4\right)\cdot 53^{9} +O(53^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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