Properties

Label 4.40428032.12t34.b.a
Dimension $4$
Group $C_3^2:D_4$
Conductor $40428032$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $4$
Group: $C_3^2:D_4$
Conductor: \(40428032\)\(\medspace = 2^{9} \cdot 281^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.143872.1
Galois orbit size: $1$
Smallest permutation container: 12T34
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $\SOPlus(4,2)$
Projective stem field: Galois closure of 6.0.143872.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + 3x^{4} - 2x^{3} + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \( x^{2} + 16x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 14 a + 4 + \left(15 a + 4\right)\cdot 17 + \left(5 a + 1\right)\cdot 17^{2} + \left(14 a + 14\right)\cdot 17^{3} + \left(7 a + 2\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 a + 1 + \left(a + 6\right)\cdot 17 + \left(11 a + 8\right)\cdot 17^{2} + \left(2 a + 5\right)\cdot 17^{3} + \left(9 a + 13\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 11 a + \left(4 a + 10\right)\cdot 17 + \left(8 a + 15\right)\cdot 17^{2} + \left(3 a + 3\right)\cdot 17^{3} + \left(3 a + 13\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 11 + \left(12 a + 3\right)\cdot 17 + \left(8 a + 2\right)\cdot 17^{2} + \left(13 a + 16\right)\cdot 17^{3} + \left(13 a + 12\right)\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 14 + 4\cdot 17 + 11\cdot 17^{2} + 14\cdot 17^{3} + 16\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 6 + 5\cdot 17 + 12\cdot 17^{2} + 13\cdot 17^{3} + 8\cdot 17^{4} +O(17^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.