Properties

Label 2.2880.8t17.a.a
Dimension $2$
Group $C_4\wr C_2$
Conductor $2880$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $C_4\wr C_2$
Conductor: \(2880\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 5 \)
Artin stem field: 8.0.23887872000.1
Galois orbit size: $2$
Smallest permutation container: $C_4\wr C_2$
Parity: odd
Determinant: 1.5.4t1.a.a
Projective image: $D_4$
Projective stem field: 4.2.18000.1

Defining polynomial

$f(x)$$=$\(x^{8} - 6 x^{4} + 45\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.

Roots:
$r_{ 1 }$ $=$ \( 2 + 3\cdot 41 + 6\cdot 41^{2} + 33\cdot 41^{3} + 33\cdot 41^{4} + 20\cdot 41^{5} + 24\cdot 41^{6} + 34\cdot 41^{7} + 15\cdot 41^{8} + 26\cdot 41^{9} +O(41^{10})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 12 + 33\cdot 41 + 33\cdot 41^{2} + 29\cdot 41^{3} + 19\cdot 41^{4} + 17\cdot 41^{5} + 27\cdot 41^{6} + 37\cdot 41^{7} + 10\cdot 41^{8} + 19\cdot 41^{9} +O(41^{10})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 + 2\cdot 41 + 13\cdot 41^{2} + 40\cdot 41^{3} + 7\cdot 41^{4} + 25\cdot 41^{5} + 24\cdot 41^{6} + 8\cdot 41^{7} + 21\cdot 41^{8} + 14\cdot 41^{9} +O(41^{10})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 18 + 4\cdot 41 + 27\cdot 41^{2} + 8\cdot 41^{3} + 22\cdot 41^{4} + 20\cdot 41^{5} + 34\cdot 41^{6} + 8\cdot 41^{7} + 33\cdot 41^{8} + 3\cdot 41^{9} +O(41^{10})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 23 + 36\cdot 41 + 13\cdot 41^{2} + 32\cdot 41^{3} + 18\cdot 41^{4} + 20\cdot 41^{5} + 6\cdot 41^{6} + 32\cdot 41^{7} + 7\cdot 41^{8} + 37\cdot 41^{9} +O(41^{10})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 26 + 38\cdot 41 + 27\cdot 41^{2} + 33\cdot 41^{4} + 15\cdot 41^{5} + 16\cdot 41^{6} + 32\cdot 41^{7} + 19\cdot 41^{8} + 26\cdot 41^{9} +O(41^{10})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 29 + 7\cdot 41 + 7\cdot 41^{2} + 11\cdot 41^{3} + 21\cdot 41^{4} + 23\cdot 41^{5} + 13\cdot 41^{6} + 3\cdot 41^{7} + 30\cdot 41^{8} + 21\cdot 41^{9} +O(41^{10})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 39 + 37\cdot 41 + 34\cdot 41^{2} + 7\cdot 41^{3} + 7\cdot 41^{4} + 20\cdot 41^{5} + 16\cdot 41^{6} + 6\cdot 41^{7} + 25\cdot 41^{8} + 14\cdot 41^{9} +O(41^{10})\)  Toggle raw display

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

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

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$$2$$(1,8)(4,5)$$0$
$4$$2$$(1,6)(2,4)(3,8)(5,7)$$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$$4$$(1,5,8,4)(2,3,7,6)$$0$
$2$$4$$(1,5,8,4)$$-\zeta_{4} + 1$
$2$$4$$(1,4,8,5)$$\zeta_{4} + 1$
$2$$4$$(1,8)(2,3,7,6)(4,5)$$\zeta_{4} - 1$
$2$$4$$(1,8)(2,6,7,3)(4,5)$$-\zeta_{4} - 1$
$4$$4$$(1,6,8,3)(2,5,7,4)$$0$
$4$$8$$(1,6,5,7,8,3,4,2)$$0$
$4$$8$$(1,7,4,6,8,2,5,3)$$0$

The blue line marks the conjugacy class containing complex conjugation.