Properties

Label 4.6642432.12t34.d.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $6642432$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(6642432\)\(\medspace = 2^{8} \cdot 3^{3} \cdot 31^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.53568.1
Galois orbit size: $1$
Smallest permutation container: 12T34
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $\SOPlus(4,2)$
Projective stem field: Galois closure of 6.0.53568.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + 2x^{4} + 2x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 28 + 29\cdot 43 + 15\cdot 43^{2} + 16\cdot 43^{3} + 4\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 2 a + 27 + \left(29 a + 8\right)\cdot 43 + \left(33 a + 27\right)\cdot 43^{2} + \left(21 a + 13\right)\cdot 43^{3} + \left(38 a + 2\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 41 a + 9 + \left(39 a + 7\right)\cdot 43 + \left(37 a + 36\right)\cdot 43^{2} + \left(25 a + 40\right)\cdot 43^{3} + \left(33 a + 36\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 41 a + 29 + \left(13 a + 35\right)\cdot 43 + \left(9 a + 31\right)\cdot 43^{2} + \left(21 a + 1\right)\cdot 43^{3} + \left(4 a + 19\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 31 + 41\cdot 43 + 26\cdot 43^{2} + 27\cdot 43^{3} + 21\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 a + 7 + \left(3 a + 6\right)\cdot 43 + \left(5 a + 34\right)\cdot 43^{2} + \left(17 a + 28\right)\cdot 43^{3} + \left(9 a + 1\right)\cdot 43^{4} +O(43^{5})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,2)(3,4)(5,6)$$-2$
$6$$2$$(3,6)$$0$
$9$$2$$(3,6)(4,5)$$0$
$4$$3$$(1,3,6)$$-2$
$4$$3$$(1,3,6)(2,4,5)$$1$
$18$$4$$(1,2)(3,5,6,4)$$0$
$12$$6$$(1,4,3,5,6,2)$$1$
$12$$6$$(2,4,5)(3,6)$$0$

The blue line marks the conjugacy class containing complex conjugation.