Properties

Label 2.2e3_17e2.8t7.1c1
Dimension 2
Group $C_8:C_2$
Conductor $ 2^{3} \cdot 17^{2}$
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$C_8:C_2$
Conductor:$2312= 2^{3} \cdot 17^{2} $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 5 x^{6} - 25 x^{5} + 42 x^{4} - 33 x^{3} + 3 x^{2} + 10 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_8:C_2$
Parity: Odd
Determinant: 1.2e3_17.4t1.2c2

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 8.
Roots: \[ \begin{aligned} r_{ 1 } &= 27 + 11\cdot 89 + 80\cdot 89^{2} + 27\cdot 89^{3} + 69\cdot 89^{4} + 77\cdot 89^{5} + 70\cdot 89^{6} + 88\cdot 89^{7} +O\left(89^{ 8 }\right) \\ r_{ 2 } &= 42 + 30\cdot 89 + 84\cdot 89^{2} + 25\cdot 89^{3} + 47\cdot 89^{4} + 32\cdot 89^{5} + 3\cdot 89^{6} + 35\cdot 89^{7} +O\left(89^{ 8 }\right) \\ r_{ 3 } &= 51 + 59\cdot 89 + 65\cdot 89^{2} + 69\cdot 89^{3} + 54\cdot 89^{4} + 68\cdot 89^{5} + 10\cdot 89^{6} + 56\cdot 89^{7} +O\left(89^{ 8 }\right) \\ r_{ 4 } &= 56 + 32\cdot 89 + 31\cdot 89^{2} + 40\cdot 89^{3} + 38\cdot 89^{4} + 38\cdot 89^{5} + 59\cdot 89^{6} + 62\cdot 89^{7} +O\left(89^{ 8 }\right) \\ r_{ 5 } &= 58 + 24\cdot 89 + 34\cdot 89^{2} + 49\cdot 89^{3} + 75\cdot 89^{4} + 43\cdot 89^{5} + 58\cdot 89^{6} + 38\cdot 89^{7} +O\left(89^{ 8 }\right) \\ r_{ 6 } &= 63 + 28\cdot 89 + 46\cdot 89^{2} + 72\cdot 89^{3} + 51\cdot 89^{4} + 37\cdot 89^{5} + 4\cdot 89^{6} + 61\cdot 89^{7} +O\left(89^{ 8 }\right) \\ r_{ 7 } &= 66 + 45\cdot 89 + 2\cdot 89^{2} + 65\cdot 89^{3} + 81\cdot 89^{4} + 28\cdot 89^{5} + 67\cdot 89^{6} + 33\cdot 89^{7} +O\left(89^{ 8 }\right) \\ r_{ 8 } &= 85 + 33\cdot 89 + 11\cdot 89^{2} + 5\cdot 89^{3} + 26\cdot 89^{4} + 28\cdot 89^{5} + 81\cdot 89^{6} + 68\cdot 89^{7} +O\left(89^{ 8 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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