Properties

Label 2.1600.8t17.a.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $1600$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(1600\)\(\medspace = 2^{6} \cdot 5^{2} \)
Artin stem field: Galois closure of 8.0.32768000.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.5.4t1.a.b
Projective image: $D_4$
Projective stem field: Galois closure of 4.2.2000.1

Defining polynomial

$f(x)$$=$ \( x^{8} + 2x^{4} + 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 7.

Roots:
$r_{ 1 }$ $=$ \( 5 + 80\cdot 101 + 6\cdot 101^{2} + 81\cdot 101^{3} + 82\cdot 101^{4} + 64\cdot 101^{5} + 26\cdot 101^{6} +O(101^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 27 + 93\cdot 101 + 60\cdot 101^{2} + 4\cdot 101^{3} + 88\cdot 101^{4} + 2\cdot 101^{5} + 33\cdot 101^{6} +O(101^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 33 + 43\cdot 101 + 60\cdot 101^{2} + 10\cdot 101^{3} + 95\cdot 101^{4} + 97\cdot 101^{5} + 80\cdot 101^{6} +O(101^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 50 + 17\cdot 101 + 7\cdot 101^{2} + 62\cdot 101^{3} + 97\cdot 101^{4} + 64\cdot 101^{5} + 22\cdot 101^{6} +O(101^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 51 + 83\cdot 101 + 93\cdot 101^{2} + 38\cdot 101^{3} + 3\cdot 101^{4} + 36\cdot 101^{5} + 78\cdot 101^{6} +O(101^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 68 + 57\cdot 101 + 40\cdot 101^{2} + 90\cdot 101^{3} + 5\cdot 101^{4} + 3\cdot 101^{5} + 20\cdot 101^{6} +O(101^{7})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 74 + 7\cdot 101 + 40\cdot 101^{2} + 96\cdot 101^{3} + 12\cdot 101^{4} + 98\cdot 101^{5} + 67\cdot 101^{6} +O(101^{7})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 96 + 20\cdot 101 + 94\cdot 101^{2} + 19\cdot 101^{3} + 18\cdot 101^{4} + 36\cdot 101^{5} + 74\cdot 101^{6} +O(101^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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