Properties

Label 2.2e4_7.8t17.1c2
Dimension 2
Group $C_4\wr C_2$
Conductor $ 2^{4} \cdot 7 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:$112= 2^{4} \cdot 7 $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 6 x^{6} - 8 x^{5} + 10 x^{4} - 9 x^{3} + 6 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4\wr C_2$
Parity: Odd
Determinant: 1.2e4_7.4t1.2c2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 239 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 30 + 93\cdot 239 + 155\cdot 239^{2} + 152\cdot 239^{3} + 154\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 92 + 154\cdot 239 + 26\cdot 239^{2} + 233\cdot 239^{3} + 21\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 122 + 203\cdot 239 + 211\cdot 239^{2} + 50\cdot 239^{3} + 166\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 126 + 68\cdot 239 + 25\cdot 239^{2} + 86\cdot 239^{3} + 60\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 187 + 73\cdot 239 + 178\cdot 239^{2} + 232\cdot 239^{3} + 208\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 194 + 202\cdot 239 + 87\cdot 239^{2} + 65\cdot 239^{3} + 69\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 211 + 85\cdot 239 + 207\cdot 239^{2} + 116\cdot 239^{3} + 137\cdot 239^{4} +O\left(239^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 236 + 73\cdot 239 + 63\cdot 239^{2} + 18\cdot 239^{3} + 137\cdot 239^{4} +O\left(239^{ 5 }\right)$

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

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

Character values on conjugacy classes

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