Properties

Label 2.2e2_17.8t17.1c1
Dimension 2
Group $C_4\wr C_2$
Conductor $ 2^{2} \cdot 17 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$C_4\wr C_2$
Conductor:$68= 2^{2} \cdot 17 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} + 4 x^{5} - 4 x^{4} + 3 x^{2} - 2 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_4\wr C_2$
Parity: Odd
Determinant: 1.2e2_17.4t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 23 + 19\cdot 149 + 46\cdot 149^{2} + 81\cdot 149^{3} + 103\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 61 + 34\cdot 149 + 129\cdot 149^{2} + 79\cdot 149^{3} + 110\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 81 + 8\cdot 149 + 91\cdot 149^{2} + 83\cdot 149^{3} + 71\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 98 + 18\cdot 149 + 67\cdot 149^{3} + 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 100 + 63\cdot 149 + 100\cdot 149^{2} + 70\cdot 149^{3} + 78\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 122 + 35\cdot 149 + 36\cdot 149^{2} + 22\cdot 149^{3} + 101\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 124 + 114\cdot 149 + 49\cdot 149^{2} + 142\cdot 149^{3} + 36\cdot 149^{4} +O\left(149^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 138 + 2\cdot 149 + 143\cdot 149^{2} + 48\cdot 149^{3} + 92\cdot 149^{4} +O\left(149^{ 5 }\right)$

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

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

Character values on conjugacy classes

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