Properties

Label 4.41e3_257.8t44.1c1
Dimension 4
Group $C_2 \wr S_4$
Conductor $ 41^{3} \cdot 257 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_2 \wr S_4$
Conductor:$17712697= 41^{3} \cdot 257 $
Artin number field: Splitting field of $f=x^{8} - x^{7} - x^{6} + x^{5} + x^{4} - x^{3} - x^{2} + x + 1$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_2 \wr S_4$
Parity: Even
Determinant: 1.41_257.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 23.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots: \[ \begin{aligned} r_{ 1 } &= 4125732070849904116955137095394880923 a + 108535691143585843174946962314492440731 +O\left(47^{ 23 }\right) \\ r_{ 2 } &= -4125732070849904116955137095394880923 a - 77122252044659842088034206978277200804 +O\left(47^{ 23 }\right) \\ r_{ 3 } &= 138459048631165464992074476883657365752 a + 61817149605088419169667164777055829502 +O\left(47^{ 23 }\right) \\ r_{ 4 } &= -138459048631165464992074476883657365752 a - 136719279473980099836979068573881549855 +O\left(47^{ 23 }\right) \\ r_{ 5 } &= 112425752280088062286582164550103345723 a - 52712332848855451861854777879647583767 +O\left(47^{ 23 }\right) \\ r_{ 6 } &= -112425752280088062286582164550103345723 a + 58518036824362378644838011434539641112 +O\left(47^{ 23 }\right) \\ r_{ 7 } &= 70856290574570332719634952915386126707 +O\left(47^{ 23 }\right) \\ r_{ 8 } &= -33173303780111579922219038009667703625 +O\left(47^{ 23 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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