Properties

Label 2.384.4t3.f.a
Dimension 2
Group $D_4$
Conductor $ 2^{7} \cdot 3 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_4$
Conductor:$384= 2^{7} \cdot 3 $
Artin number field: Splitting field of 8.0.339738624.6 defined by $f= x^{8} - 4 x^{7} + 8 x^{6} - 4 x^{5} - 4 x^{4} - 4 x^{3} + 16 x^{2} - 12 x + 9 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $D_{4}$
Parity: Odd
Determinant: 1.24.2t1.b.a
Projective image: $C_2^2$
Projective field: Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 8 + 13\cdot 97 + 50\cdot 97^{2} + 89\cdot 97^{3} + 64\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 22 + 20\cdot 97 + 47\cdot 97^{2} + 68\cdot 97^{3} + 67\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 25 + 50\cdot 97 + 55\cdot 97^{2} + 32\cdot 97^{3} + 48\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 39 + 57\cdot 97 + 52\cdot 97^{2} + 11\cdot 97^{3} + 51\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 63 + 80\cdot 97 + 16\cdot 97^{2} + 72\cdot 97^{3} + 57\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 69 + 5\cdot 97 + 69\cdot 97^{2} + 77\cdot 97^{3} + 36\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 80 + 20\cdot 97 + 22\cdot 97^{2} + 15\cdot 97^{3} + 41\cdot 97^{4} +O\left(97^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 86 + 42\cdot 97 + 74\cdot 97^{2} + 20\cdot 97^{3} + 20\cdot 97^{4} +O\left(97^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,5)(2,6)(3,7)(4,8)$$-2$
$2$$2$$(1,2)(3,4)(5,6)(7,8)$$0$
$2$$2$$(1,3)(2,8)(4,6)(5,7)$$0$
$2$$4$$(1,8,5,4)(2,3,6,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.