Properties

Label 6.848081520125.8t33.b.a
Dimension $6$
Group $C_2^4:C_6$
Conductor $848081520125$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $6$
Group: $C_2^4:C_6$
Conductor: \(848081520125\)\(\medspace = 5^{3} \cdot 7^{4} \cdot 41^{4} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 8.4.2522550625.1
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 8.4.2522550625.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 3x^{7} - x^{6} + 9x^{5} - 11x^{4} + 11x^{3} - 13x^{2} + 21x - 9 \) Copy content Toggle raw display .

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:
$r_{ 1 }$ $=$ \( 2 + 11 + 3\cdot 11^{2} + 9\cdot 11^{3} + 2\cdot 11^{4} + 5\cdot 11^{6} + 3\cdot 11^{7} + 3\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a^{2} + 5 a + 8 + \left(8 a^{2} + 5 a + 9\right)\cdot 11 + \left(8 a^{2} + 8 a + 9\right)\cdot 11^{2} + \left(5 a + 4\right)\cdot 11^{3} + \left(6 a^{2} + 9 a + 9\right)\cdot 11^{4} + \left(5 a^{2} + 1\right)\cdot 11^{5} + \left(5 a^{2} + 2 a + 10\right)\cdot 11^{6} + \left(a^{2} + a + 6\right)\cdot 11^{7} + \left(2 a^{2} + 7 a + 1\right)\cdot 11^{8} + \left(4 a^{2} + 7 a\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 2 a^{2} + 7 a + \left(2 a^{2} + 3 a\right)\cdot 11 + \left(9 a^{2} + 2 a + 4\right)\cdot 11^{2} + \left(8 a^{2} + 4 a + 2\right)\cdot 11^{3} + \left(9 a^{2} + 6 a + 10\right)\cdot 11^{4} + \left(10 a^{2} + 1\right)\cdot 11^{5} + \left(5 a^{2} + 7 a + 7\right)\cdot 11^{6} + \left(5 a^{2} + 10 a + 6\right)\cdot 11^{7} + \left(8 a^{2} + 2 a + 1\right)\cdot 11^{8} + \left(3 a^{2} + 4 a + 4\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 1 + \left(6 a^{2} + 10 a + 6\right)\cdot 11 + \left(5 a^{2} + 8 a + 5\right)\cdot 11^{2} + \left(2 a^{2} + 7 a + 3\right)\cdot 11^{3} + \left(2 a^{2} + 4 a + 4\right)\cdot 11^{4} + \left(4 a^{2} + 3 a + 7\right)\cdot 11^{5} + \left(7 a^{2} + 4 a + 1\right)\cdot 11^{6} + \left(4 a^{2} + 3 a\right)\cdot 11^{7} + \left(7 a^{2} + a + 5\right)\cdot 11^{8} + \left(6 a^{2} + 8 a + 3\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a^{2} + a + 5 + \left(7 a^{2} + 5 a + 10\right)\cdot 11 + \left(9 a^{2} + 7 a\right)\cdot 11^{2} + \left(6 a^{2} + 10 a + 7\right)\cdot 11^{3} + \left(10 a^{2} + 7 a + 7\right)\cdot 11^{4} + \left(3 a^{2} + 5 a + 3\right)\cdot 11^{5} + \left(2 a^{2} + 3 a + 2\right)\cdot 11^{6} + \left(a^{2} + 4 a + 8\right)\cdot 11^{7} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{8} + \left(10 a^{2} + 4 a + 9\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a^{2} + 3 a + 5 + \left(7 a^{2} + 6 a\right)\cdot 11 + \left(7 a^{2} + 4 a + 1\right)\cdot 11^{2} + \left(7 a^{2} + 8 a + 3\right)\cdot 11^{3} + \left(2 a^{2} + 7 a + 1\right)\cdot 11^{4} + \left(a^{2} + 6 a + 7\right)\cdot 11^{5} + \left(9 a^{2} + 4 a + 7\right)\cdot 11^{6} + \left(4 a^{2} + 6 a + 7\right)\cdot 11^{7} + \left(a^{2} + 2 a\right)\cdot 11^{8} + \left(6 a + 2\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 6 + 2\cdot 11 + 5\cdot 11^{2} + 7\cdot 11^{3} + 11^{4} + 10\cdot 11^{5} + 10\cdot 11^{6} + 5\cdot 11^{7} + 7\cdot 11^{8} + 3\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 6 a^{2} + 3 a + 9 + \left(a^{2} + 2 a + 2\right)\cdot 11 + \left(3 a^{2} + a + 3\right)\cdot 11^{2} + \left(6 a^{2} + 7 a + 6\right)\cdot 11^{3} + \left(a^{2} + 7 a + 6\right)\cdot 11^{4} + \left(7 a^{2} + 4 a\right)\cdot 11^{5} + \left(2 a^{2} + 10\right)\cdot 11^{6} + \left(4 a^{2} + 7 a + 4\right)\cdot 11^{7} + \left(4 a^{2} + 5 a + 3\right)\cdot 11^{8} + \left(7 a^{2} + 2 a + 5\right)\cdot 11^{9} +O(11^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.