Properties

Label 2.491.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 491 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$491 $
Artin number field: Splitting field of $f=x^{8} - 2 x^{7} - x^{5} + 7 x^{3} + x - 9$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.491.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$
Roots: \[ \begin{aligned} r_{ 1 } &= -65501842 a - 21851162 +O\left(47^{ 5 }\right) \\ r_{ 2 } &= 34869023 a - 20785166 +O\left(47^{ 5 }\right) \\ r_{ 3 } &= 65501842 a - 55753363 +O\left(47^{ 5 }\right) \\ r_{ 4 } &= -34869023 a + 15523848 +O\left(47^{ 5 }\right) \\ r_{ 5 } &= 16243678 a + 29109024 +O\left(47^{ 5 }\right) \\ r_{ 6 } &= -36573485 +O\left(47^{ 5 }\right) \\ r_{ 7 } &= -16243678 a - 13821465 +O\left(47^{ 5 }\right) \\ r_{ 8 } &= 104151771 +O\left(47^{ 5 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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