Properties

Label 2.563.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 563 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$563 $
Artin number field: Splitting field of $f=x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 4 x^{4} + 12 x^{3} - 7 x^{2} + 2 x - 9$ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.563.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $x^{2} + 24 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= 50226017 a - 175382018 +O\left(29^{ 6 }\right) \\ r_{ 2 } &= 7516107 a + 204686057 +O\left(29^{ 6 }\right) \\ r_{ 3 } &= 165321053 +O\left(29^{ 6 }\right) \\ r_{ 4 } &= -7516107 a + 24299489 +O\left(29^{ 6 }\right) \\ r_{ 5 } &= -27484451 a + 262301390 +O\left(29^{ 6 }\right) \\ r_{ 6 } &= 27484451 a - 267718428 +O\left(29^{ 6 }\right) \\ r_{ 7 } &= -22347757 +O\left(29^{ 6 }\right) \\ r_{ 8 } &= -50226017 a - 191159784 +O\left(29^{ 6 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

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