Properties

Label 2.1792.8t8.b.a
Dimension $2$
Group $QD_{16}$
Conductor $1792$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $QD_{16}$
Conductor: \(1792\)\(\medspace = 2^{8} \cdot 7 \)
Artin stem field: 8.2.5754585088.3
Galois orbit size: $2$
Smallest permutation container: $QD_{16}$
Parity: odd
Determinant: 1.7.2t1.a.a
Projective image: $D_4$
Projective stem field: 4.0.1568.1

Defining polynomial

$f(x)$$=$\(x^{8} - 4 x^{6} + 4 x^{4} + 4 x^{2} - 7\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 71 }$ to precision 8.

Roots:
$r_{ 1 }$ $=$ \( 5 + 35\cdot 71 + 22\cdot 71^{2} + 64\cdot 71^{3} + 68\cdot 71^{4} + 71^{5} + 69\cdot 71^{6} + 27\cdot 71^{7} +O(71^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 6 + 60\cdot 71 + 71^{2} + 29\cdot 71^{3} + 48\cdot 71^{4} + 41\cdot 71^{5} + 58\cdot 71^{6} + 16\cdot 71^{7} +O(71^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 18 + 4\cdot 71 + 70\cdot 71^{2} + 38\cdot 71^{3} + 62\cdot 71^{4} + 40\cdot 71^{5} + 18\cdot 71^{6} + 60\cdot 71^{7} +O(71^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 20 + 67\cdot 71 + 64\cdot 71^{2} + 37\cdot 71^{3} + 50\cdot 71^{4} + 6\cdot 71^{5} + 49\cdot 71^{6} + 24\cdot 71^{7} +O(71^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 51 + 3\cdot 71 + 6\cdot 71^{2} + 33\cdot 71^{3} + 20\cdot 71^{4} + 64\cdot 71^{5} + 21\cdot 71^{6} + 46\cdot 71^{7} +O(71^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 53 + 66\cdot 71 + 32\cdot 71^{3} + 8\cdot 71^{4} + 30\cdot 71^{5} + 52\cdot 71^{6} + 10\cdot 71^{7} +O(71^{8})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 65 + 10\cdot 71 + 69\cdot 71^{2} + 41\cdot 71^{3} + 22\cdot 71^{4} + 29\cdot 71^{5} + 12\cdot 71^{6} + 54\cdot 71^{7} +O(71^{8})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 66 + 35\cdot 71 + 48\cdot 71^{2} + 6\cdot 71^{3} + 2\cdot 71^{4} + 69\cdot 71^{5} + 71^{6} + 43\cdot 71^{7} +O(71^{8})\)  Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.