Properties

Label 2.2e7_7.8t6.1c1
Dimension 2
Group $D_{8}$
Conductor $ 2^{7} \cdot 7 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:$896= 2^{7} \cdot 7 $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 4 x^{6} - 8 x^{5} + 14 x^{4} - 8 x^{3} + 4 x^{2} - 8 x - 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $D_{8}$
Parity: Odd
Determinant: 1.2e3_7.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 6.
Roots: \[ \begin{aligned} r_{ 1 } &= 15 + 36\cdot 191 + 75\cdot 191^{2} + 188\cdot 191^{3} + 2\cdot 191^{4} + 130\cdot 191^{5} +O\left(191^{ 6 }\right) \\ r_{ 2 } &= 31 + 106\cdot 191 + 136\cdot 191^{2} + 165\cdot 191^{3} + 122\cdot 191^{4} + 97\cdot 191^{5} +O\left(191^{ 6 }\right) \\ r_{ 3 } &= 33 + 146\cdot 191 + 90\cdot 191^{2} + 42\cdot 191^{3} + 73\cdot 191^{4} + 59\cdot 191^{5} +O\left(191^{ 6 }\right) \\ r_{ 4 } &= 52 + 187\cdot 191 + 59\cdot 191^{2} + 53\cdot 191^{3} + 81\cdot 191^{4} + 105\cdot 191^{5} +O\left(191^{ 6 }\right) \\ r_{ 5 } &= 69 + 6\cdot 191 + 81\cdot 191^{2} + 116\cdot 191^{3} + 52\cdot 191^{4} + 41\cdot 191^{5} +O\left(191^{ 6 }\right) \\ r_{ 6 } &= 95 + 52\cdot 191 + 110\cdot 191^{2} + 165\cdot 191^{3} + 174\cdot 191^{4} + 48\cdot 191^{5} +O\left(191^{ 6 }\right) \\ r_{ 7 } &= 120 + 45\cdot 191 + 167\cdot 191^{2} + 76\cdot 191^{3} + 178\cdot 191^{4} + 19\cdot 191^{5} +O\left(191^{ 6 }\right) \\ r_{ 8 } &= 162 + 183\cdot 191 + 42\cdot 191^{2} + 146\cdot 191^{3} + 77\cdot 191^{4} + 70\cdot 191^{5} +O\left(191^{ 6 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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