Properties

Label 6.296065125.8t33.b.a
Dimension $6$
Group $C_2^4:C_6$
Conductor $296065125$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $C_2^4:C_6$
Conductor: \(296065125\)\(\medspace = 3^{8} \cdot 5^{3} \cdot 19^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.0.1480325625.1
Galois orbit size: $1$
Smallest permutation container: $C_2^4:C_6$
Parity: even
Determinant: 1.5.2t1.a.a
Projective image: $C_2^3:A_4$
Projective stem field: Galois closure of 8.0.1480325625.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 3x^{7} + x^{6} + 7x^{5} - 3x^{4} - 5x^{3} + 11x^{2} + 15x + 5 \) 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 $\mathbb{Q}_{ 29 }$: \( x^{3} + 2x + 27 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 1 + 16\cdot 29 + 10\cdot 29^{2} + 4\cdot 29^{3} + 2\cdot 29^{4} + 20\cdot 29^{5} + 3\cdot 29^{6} + 11\cdot 29^{8} + 22\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 12 a^{2} + 17 a + 22 + \left(3 a^{2} + 27 a + 27\right)\cdot 29 + \left(5 a^{2} + 2 a + 15\right)\cdot 29^{2} + \left(24 a + 12\right)\cdot 29^{3} + \left(22 a^{2} + 3 a + 2\right)\cdot 29^{4} + \left(19 a^{2} + 26 a + 13\right)\cdot 29^{5} + \left(2 a^{2} + 17 a + 1\right)\cdot 29^{6} + \left(9 a^{2} + 13 a + 17\right)\cdot 29^{7} + \left(15 a^{2} + 23 a\right)\cdot 29^{8} + \left(19 a^{2} + 10 a + 2\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 4 a^{2} + 19 a + 21 + \left(2 a^{2} + 5 a + 6\right)\cdot 29 + \left(a^{2} + 9 a + 20\right)\cdot 29^{2} + \left(17 a^{2} + 19 a + 15\right)\cdot 29^{3} + \left(17 a^{2} + 18 a + 25\right)\cdot 29^{4} + \left(3 a^{2} + 25 a + 10\right)\cdot 29^{5} + \left(2 a^{2} + 5 a + 10\right)\cdot 29^{6} + \left(2 a^{2} + 17\right)\cdot 29^{7} + \left(28 a^{2} + 27 a + 17\right)\cdot 29^{8} + \left(6 a^{2} + 10 a + 4\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 10 a^{2} + 15 a + 2 + \left(2 a^{2} + 24 a + 10\right)\cdot 29 + \left(22 a^{2} + 27 a + 24\right)\cdot 29^{2} + \left(a^{2} + 18 a + 10\right)\cdot 29^{3} + \left(15 a^{2} + 16 a + 14\right)\cdot 29^{4} + \left(2 a^{2} + 27 a + 15\right)\cdot 29^{5} + \left(a^{2} + 7 a\right)\cdot 29^{6} + \left(11 a^{2} + 18 a + 5\right)\cdot 29^{7} + \left(20 a^{2} + 22 a + 21\right)\cdot 29^{8} + \left(5 a^{2} + 10 a + 6\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 13 a^{2} + 22 a + 4 + \left(23 a^{2} + 24 a + 6\right)\cdot 29 + \left(22 a^{2} + 16 a + 20\right)\cdot 29^{2} + \left(11 a^{2} + 14 a + 8\right)\cdot 29^{3} + \left(18 a^{2} + 6 a + 7\right)\cdot 29^{4} + \left(5 a^{2} + 6 a + 23\right)\cdot 29^{5} + \left(24 a^{2} + 5 a + 10\right)\cdot 29^{6} + \left(17 a^{2} + 15 a + 9\right)\cdot 29^{7} + \left(14 a^{2} + 7 a + 9\right)\cdot 29^{8} + \left(2 a^{2} + 7 a + 8\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 24 a^{2} + 15 a + 11 + \left(3 a^{2} + 4 a + 2\right)\cdot 29 + \left(2 a^{2} + 18 a + 17\right)\cdot 29^{2} + \left(25 a^{2} + 17 a + 22\right)\cdot 29^{3} + \left(16 a^{2} + a + 16\right)\cdot 29^{4} + \left(7 a^{2} + 2 a + 12\right)\cdot 29^{5} + \left(23 a^{2} + 13 a + 20\right)\cdot 29^{6} + \left(16 a^{2} + 8 a + 12\right)\cdot 29^{7} + \left(7 a^{2} + 2 a + 23\right)\cdot 29^{8} + \left(23 a + 18\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 18 + 9\cdot 29 + 6\cdot 29^{2} + 20\cdot 29^{3} + 8\cdot 29^{4} + 12\cdot 29^{5} + 5\cdot 29^{6} + 14\cdot 29^{7} + 8\cdot 29^{8} + 23\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 24 a^{2} + 28 a + 11 + \left(22 a^{2} + 28 a + 8\right)\cdot 29 + \left(4 a^{2} + 11 a + 1\right)\cdot 29^{2} + \left(2 a^{2} + 21 a + 21\right)\cdot 29^{3} + \left(26 a^{2} + 10 a + 9\right)\cdot 29^{4} + \left(18 a^{2} + 28 a + 8\right)\cdot 29^{5} + \left(4 a^{2} + 7 a + 5\right)\cdot 29^{6} + \left(a^{2} + 2 a + 11\right)\cdot 29^{7} + \left(a^{2} + 4 a + 24\right)\cdot 29^{8} + \left(23 a^{2} + 24 a\right)\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$6$
$3$$2$$(1,6)(2,5)(3,7)(4,8)$$-2$
$4$$2$$(1,7)(2,4)(3,6)(5,8)$$0$
$6$$2$$(1,4)(2,3)(5,7)(6,8)$$-2$
$6$$2$$(1,6)(4,8)$$2$
$16$$3$$(1,8,4)(3,5,7)$$0$
$16$$3$$(1,4,8)(3,7,5)$$0$
$12$$4$$(1,7,6,3)(2,8,5,4)$$0$
$16$$6$$(1,5)(2,4,7,6,3,8)$$0$
$16$$6$$(1,5)(2,8,3,6,7,4)$$0$

The blue line marks the conjugacy class containing complex conjugation.