Properties

Label 4.5_11e2_29e2.8t29.3c1
Dimension 4
Group $(((C_4 \times C_2): C_2):C_2):C_2$
Conductor $ 5 \cdot 11^{2} \cdot 29^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$4$
Group:$(((C_4 \times C_2): C_2):C_2):C_2$
Conductor:$508805= 5 \cdot 11^{2} \cdot 29^{2} $
Artin number field: Splitting field of $f=x^{8} - x^{7} + 3 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 16 x^{2} - 20 x + 9$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $(((C_4 \times C_2): C_2):C_2):C_2$
Parity: Even
Determinant: 1.5.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 929 }$ to precision 7.
Roots: \[ \begin{aligned} r_{ 1 } &= 160591522267966665177 +O\left(929^{ 7 }\right) \\ r_{ 2 } &= 179361339923196886811 +O\left(929^{ 7 }\right) \\ r_{ 3 } &= 26508495031979191456 +O\left(929^{ 7 }\right) \\ r_{ 4 } &= 160861733878600930251 +O\left(929^{ 7 }\right) \\ r_{ 5 } &= -145398331870544416033 +O\left(929^{ 7 }\right) \\ r_{ 6 } &= 163474160924526753914 +O\left(929^{ 7 }\right) \\ r_{ 7 } &= 283743415898996497344 +O\left(929^{ 7 }\right) \\ r_{ 8 } &= -231955813429226220310 +O\left(929^{ 7 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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