Properties

Label 4.5_643.8t44.2c1
Dimension 4
Group $C_2 \wr S_4$
Conductor $ 5 \cdot 643 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_2 \wr S_4$
Conductor:$3215= 5 \cdot 643 $
Artin number field: Splitting field of $f=x^{8} - x^{7} + x^{6} - x^{4} + x^{2} - x + 1$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2 \wr S_4$
Parity: Odd
Determinant: 1.5_643.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 24.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $x^{2} + 33 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= -10879089857664740108743537254909893990 a - 5534112503528965557416556244176585763 +O\left(37^{ 24 }\right) \\ r_{ 2 } &= 16252422011727496995779574402066550784 a + 12327557674306307576142201915032718296 +O\left(37^{ 24 }\right) \\ r_{ 3 } &= 165471356817853869205937049757191855 +O\left(37^{ 24 }\right) \\ r_{ 4 } &= 10879089857664740108743537254909893990 a + 6793795909860706828199884501251924219 +O\left(37^{ 24 }\right) \\ r_{ 5 } &= 17324779277037884168275208396581583272 a - 11761952438903522657164683362425918667 +O\left(37^{ 24 }\right) \\ r_{ 6 } &= -21225407595724527973627408223101178706 +O\left(37^{ 24 }\right) \\ r_{ 7 } &= -16252422011727496995779574402066550784 a - 3979283378380966480203320353266470044 +O\left(37^{ 24 }\right) \\ r_{ 8 } &= -17324779277037884168275208396581583272 a - 20121326135640229505501091366396430150 +O\left(37^{ 24 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 8 }$ Character value
$1$$1$$()$$4$
$1$$2$$(1,8)(2,7)(3,6)(4,5)$$-4$
$4$$2$$(2,7)$$2$
$4$$2$$(1,8)(2,7)(3,6)$$-2$
$6$$2$$(2,7)(4,5)$$0$
$12$$2$$(1,3)(2,4)(5,7)(6,8)$$0$
$12$$2$$(1,4)(5,8)$$2$
$12$$2$$(1,8)(2,3)(4,5)(6,7)$$-2$
$24$$2$$(1,4)(2,7)(5,8)$$0$
$32$$3$$(1,3,2)(6,7,8)$$1$
$12$$4$$(1,3,8,6)(2,5,7,4)$$0$
$12$$4$$(1,4,8,5)$$2$
$12$$4$$(1,8)(2,6,7,3)(4,5)$$-2$
$24$$4$$(1,3)(2,5,7,4)(6,8)$$0$
$24$$4$$(1,4,8,5)(2,7)$$0$
$48$$4$$(1,4,3,2)(5,6,7,8)$$0$
$32$$6$$(1,3,2,8,6,7)$$1$
$32$$6$$(1,3,2)(4,5)(6,7,8)$$-1$
$32$$6$$(1,3,2,8,6,7)(4,5)$$-1$
$48$$8$$(1,4,3,2,8,5,6,7)$$0$
The blue line marks the conjugacy class containing complex conjugation.