Properties

Label 2.3200.8t17.b.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $3200$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(3200\)\(\medspace = 2^{7} \cdot 5^{2}\)
Artin stem field: Galois closure of 8.0.131072000.2
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.40.4t1.b.a
Projective image: $D_4$
Projective stem field: Galois closure of 4.2.8000.1

Defining polynomial

$f(x)$$=$ \( x^{8} - 6x^{6} + 12x^{4} - 10x^{2} + 5 \) Copy content Toggle raw display .

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 2 + 20\cdot 29 + 25\cdot 29^{2} + 17\cdot 29^{3} + 9\cdot 29^{4} + 9\cdot 29^{5} + 11\cdot 29^{6} + 12\cdot 29^{7} + 5\cdot 29^{8} + 9\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 23\cdot 29 + 27\cdot 29^{2} + 20\cdot 29^{3} + 28\cdot 29^{4} + 5\cdot 29^{5} + 18\cdot 29^{6} + 18\cdot 29^{7} + 11\cdot 29^{8} + 2\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 4 + 26\cdot 29 + 17\cdot 29^{2} + 22\cdot 29^{3} + 15\cdot 29^{4} + 19\cdot 29^{5} + 4\cdot 29^{6} + 25\cdot 29^{7} + 5\cdot 29^{8} +O(29^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 + 4\cdot 29 + 25\cdot 29^{2} + 7\cdot 29^{3} + 28\cdot 29^{4} + 10\cdot 29^{5} + 17\cdot 29^{6} + 10\cdot 29^{7} + 5\cdot 29^{8} +O(29^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 21 + 24\cdot 29 + 3\cdot 29^{2} + 21\cdot 29^{3} + 18\cdot 29^{5} + 11\cdot 29^{6} + 18\cdot 29^{7} + 23\cdot 29^{8} + 28\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 25 + 2\cdot 29 + 11\cdot 29^{2} + 6\cdot 29^{3} + 13\cdot 29^{4} + 9\cdot 29^{5} + 24\cdot 29^{6} + 3\cdot 29^{7} + 23\cdot 29^{8} + 28\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 26 + 5\cdot 29 + 29^{2} + 8\cdot 29^{3} + 23\cdot 29^{5} + 10\cdot 29^{6} + 10\cdot 29^{7} + 17\cdot 29^{8} + 26\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 27 + 8\cdot 29 + 3\cdot 29^{2} + 11\cdot 29^{3} + 19\cdot 29^{4} + 19\cdot 29^{5} + 17\cdot 29^{6} + 16\cdot 29^{7} + 23\cdot 29^{8} + 19\cdot 29^{9} +O(29^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.