Properties

Label 1.17.8t1.1c1
Dimension 1
Group $C_8$
Conductor $ 17 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$1$
Group:$C_8$
Conductor:$17 $
Artin number field: Splitting field of $f= x^{8} - x^{7} - 7 x^{6} + 6 x^{5} + 15 x^{4} - 10 x^{3} - 10 x^{2} + 4 x + 1 $ over $\Q$
Size of Galois orbit: 4
Smallest containing permutation representation: $C_8$
Parity: Even
Corresponding Dirichlet character: \(\chi_{17}(8,\cdot)\)

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 4 + 49\cdot 67 + 2\cdot 67^{2} + 56\cdot 67^{3} + 16\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 7 + 9\cdot 67 + 65\cdot 67^{2} + 22\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 20 + 7\cdot 67 + 12\cdot 67^{2} + 21\cdot 67^{3} + 5\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 34 + 55\cdot 67 + 10\cdot 67^{2} + 42\cdot 67^{3} + 58\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 45 + 47\cdot 67 + 37\cdot 67^{2} + 11\cdot 67^{3} + 50\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 52 + 61\cdot 67 + 57\cdot 67^{2} + 7\cdot 67^{3} + 53\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 53 + 9\cdot 67 + 56\cdot 67^{2} + 56\cdot 67^{3} + 64\cdot 67^{4} +O\left(67^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 54 + 27\cdot 67 + 25\cdot 67^{2} + 4\cdot 67^{3} + 64\cdot 67^{4} +O\left(67^{ 5 }\right)$

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

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

Character values on conjugacy classes

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