Properties

Label 6.172901784411.8t47.a.a
Dimension $6$
Group $S_4\wr C_2$
Conductor $172901784411$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $S_4\wr C_2$
Conductor: \(172901784411\)\(\medspace = 3^{6} \cdot 619^{3} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.6.321078613651227.1
Galois orbit size: $1$
Smallest permutation container: $S_4\wr C_2$
Parity: odd
Determinant: 1.619.2t1.a.a
Projective image: $S_4\wr C_2$
Projective stem field: Galois closure of 8.6.321078613651227.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 4x^{7} - 14x^{6} + 17x^{5} + 43x^{4} + 323x^{3} + 310x^{2} - 1135x - 413 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 25 a + 28 + \left(8 a + 27\right)\cdot 29 + 25\cdot 29^{2} + \left(27 a + 14\right)\cdot 29^{3} + \left(19 a + 22\right)\cdot 29^{4} + \left(21 a + 2\right)\cdot 29^{5} + \left(25 a + 14\right)\cdot 29^{6} + 7 a\cdot 29^{7} + \left(4 a + 15\right)\cdot 29^{8} + \left(11 a + 10\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 10\cdot 29 + 11\cdot 29^{2} + 4\cdot 29^{3} + 8\cdot 29^{4} + 15\cdot 29^{5} + 15\cdot 29^{6} + 25\cdot 29^{7} + 21\cdot 29^{8} + 8\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 28 a + 7 a\cdot 29 + \left(22 a + 13\right)\cdot 29^{2} + 17 a\cdot 29^{3} + \left(28 a + 9\right)\cdot 29^{4} + \left(3 a + 15\right)\cdot 29^{5} + \left(27 a + 11\right)\cdot 29^{6} + \left(28 a + 6\right)\cdot 29^{7} + \left(19 a + 15\right)\cdot 29^{8} + \left(26 a + 8\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a + 24 + \left(21 a + 11\right)\cdot 29 + 6 a\cdot 29^{2} + \left(11 a + 9\right)\cdot 29^{3} + 18\cdot 29^{4} + \left(25 a + 6\right)\cdot 29^{5} + \left(a + 27\right)\cdot 29^{6} + 7\cdot 29^{7} + \left(9 a + 28\right)\cdot 29^{8} + \left(2 a + 5\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 26 a + 19 + \left(18 a + 2\right)\cdot 29 + \left(19 a + 15\right)\cdot 29^{2} + \left(18 a + 18\right)\cdot 29^{3} + \left(5 a + 4\right)\cdot 29^{4} + \left(7 a + 21\right)\cdot 29^{5} + \left(6 a + 2\right)\cdot 29^{6} + \left(24 a + 10\right)\cdot 29^{7} + \left(8 a + 17\right)\cdot 29^{8} + \left(a + 10\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 23 + 2\cdot 29 + 24\cdot 29^{2} + 2\cdot 29^{4} + 28\cdot 29^{5} + 12\cdot 29^{6} + 13\cdot 29^{7} + 10\cdot 29^{8} + 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 4 a + 8 + \left(20 a + 18\right)\cdot 29 + \left(28 a + 18\right)\cdot 29^{2} + \left(a + 4\right)\cdot 29^{3} + \left(9 a + 8\right)\cdot 29^{4} + \left(7 a + 4\right)\cdot 29^{5} + \left(3 a + 5\right)\cdot 29^{6} + \left(21 a + 14\right)\cdot 29^{7} + \left(24 a + 28\right)\cdot 29^{8} + \left(17 a + 3\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 3 a + 4 + \left(10 a + 13\right)\cdot 29 + \left(9 a + 7\right)\cdot 29^{2} + \left(10 a + 5\right)\cdot 29^{3} + \left(23 a + 14\right)\cdot 29^{4} + \left(21 a + 22\right)\cdot 29^{5} + \left(22 a + 26\right)\cdot 29^{6} + \left(4 a + 8\right)\cdot 29^{7} + \left(20 a + 8\right)\cdot 29^{8} + \left(27 a + 8\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$6$
$6$$2$$(1,4)(3,7)$$2$
$9$$2$$(1,4)(2,6)(3,7)(5,8)$$-2$
$12$$2$$(1,3)$$4$
$24$$2$$(1,2)(3,5)(4,6)(7,8)$$0$
$36$$2$$(1,3)(2,5)$$2$
$36$$2$$(1,3)(2,6)(5,8)$$0$
$16$$3$$(3,4,7)$$3$
$64$$3$$(3,4,7)(5,6,8)$$0$
$12$$4$$(1,3,4,7)$$2$
$36$$4$$(1,3,4,7)(2,5,6,8)$$-2$
$36$$4$$(1,4)(2,5,6,8)(3,7)$$-2$
$72$$4$$(1,6,4,2)(3,8,7,5)$$0$
$72$$4$$(1,3)(2,5,6,8)$$0$
$144$$4$$(1,5,3,2)(4,6)(7,8)$$0$
$48$$6$$(2,6)(3,7,4)(5,8)$$-1$
$96$$6$$(1,3)(2,6,8)$$1$
$192$$6$$(1,2)(3,6,4,8,7,5)$$0$
$144$$8$$(1,5,3,6,4,8,7,2)$$0$
$96$$12$$(2,5,6,8)(3,4,7)$$-1$

The blue line marks the conjugacy class containing complex conjugation.