Properties

Label 2.3e2_5_19.8t6.2c1
Dimension 2
Group $D_{8}$
Conductor $ 3^{2} \cdot 5 \cdot 19 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{8}$
Conductor:$855= 3^{2} \cdot 5 \cdot 19 $
Artin number field: Splitting field of $f= x^{8} - x^{7} + 3 x^{5} + 9 x^{4} + 2 x^{3} + 8 x^{2} - 4 x + 7 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $D_{8}$
Parity: Odd
Determinant: 1.5_19.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 191 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 39 + 111\cdot 191 + 175\cdot 191^{2} + 190\cdot 191^{3} + 41\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 54 + 71\cdot 191 + 122\cdot 191^{2} + 175\cdot 191^{3} + 108\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 100 + 63\cdot 191 + 99\cdot 191^{2} + 63\cdot 191^{3} + 190\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 114 + 149\cdot 191 + 64\cdot 191^{2} + 116\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 139 + 2\cdot 191 + 128\cdot 191^{2} + 60\cdot 191^{3} + 54\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 160 + 97\cdot 191 + 7\cdot 191^{2} + 97\cdot 191^{3} + 135\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 163 + 9\cdot 191 + 105\cdot 191^{2} + 4\cdot 191^{3} + 101\cdot 191^{4} +O\left(191^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 187 + 66\cdot 191 + 61\cdot 191^{2} + 171\cdot 191^{3} + 15\cdot 191^{4} +O\left(191^{ 5 }\right)$

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

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

Character values on conjugacy classes

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