Properties

Label 2.3072.8t6.a.b
Dimension $2$
Group $D_{8}$
Conductor $3072$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{8}$
Conductor: \(3072\)\(\medspace = 2^{10} \cdot 3 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 8.0.57982058496.7
Galois orbit size: $2$
Smallest permutation container: $D_{8}$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $D_4$
Projective stem field: 4.0.6144.1

Defining polynomial

$f(x)$$=$\(x^{8} + 24 x^{4} - 32 x^{2} + 24\)  Toggle raw display.

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 7.

Roots:
$r_{ 1 }$ $=$ \( 2 + 19\cdot 67 + 46\cdot 67^{2} + 64\cdot 67^{3} + 49\cdot 67^{4} + 21\cdot 67^{5} + 61\cdot 67^{6} +O(67^{7})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 3 + 41\cdot 67 + 42\cdot 67^{2} + 3\cdot 67^{3} + 44\cdot 67^{4} + 46\cdot 67^{5} + 29\cdot 67^{6} +O(67^{7})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 10 + 34\cdot 67 + 42\cdot 67^{2} + 2\cdot 67^{3} + 62\cdot 67^{4} + 11\cdot 67^{5} + 14\cdot 67^{6} +O(67^{7})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 61\cdot 67 + 48\cdot 67^{3} + 31\cdot 67^{4} + 46\cdot 67^{5} + 15\cdot 67^{6} +O(67^{7})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 50 + 5\cdot 67 + 66\cdot 67^{2} + 18\cdot 67^{3} + 35\cdot 67^{4} + 20\cdot 67^{5} + 51\cdot 67^{6} +O(67^{7})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 57 + 32\cdot 67 + 24\cdot 67^{2} + 64\cdot 67^{3} + 4\cdot 67^{4} + 55\cdot 67^{5} + 52\cdot 67^{6} +O(67^{7})\)  Toggle raw display
$r_{ 7 }$ $=$ \( 64 + 25\cdot 67 + 24\cdot 67^{2} + 63\cdot 67^{3} + 22\cdot 67^{4} + 20\cdot 67^{5} + 37\cdot 67^{6} +O(67^{7})\)  Toggle raw display
$r_{ 8 }$ $=$ \( 65 + 47\cdot 67 + 20\cdot 67^{2} + 2\cdot 67^{3} + 17\cdot 67^{4} + 45\cdot 67^{5} + 5\cdot 67^{6} +O(67^{7})\)  Toggle raw display

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

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

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

The blue line marks the conjugacy class containing complex conjugation.