Properties

Label 2.416.8t17.a.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $416$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(416\)\(\medspace = 2^{5} \cdot 13 \)
Artin stem field: Galois closure of 8.0.143982592.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.13.4t1.a.a
Projective image: $D_4$
Projective stem field: Galois closure of 4.2.35152.1

Defining polynomial

$f(x)$$=$ \( x^{8} + 6x^{4} + 13 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.