Properties

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

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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:
$r_{ 1 }$ $=$ $ 18 a + 6 + \left(22 a + 4\right)\cdot 29 + \left(10 a + 28\right)\cdot 29^{2} + \left(13 a + 13\right)\cdot 29^{4} + \left(2 a + 20\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 26 + \left(3 a + 3\right)\cdot 29 + \left(5 a + 16\right)\cdot 29^{2} + \left(18 a + 11\right)\cdot 29^{3} + \left(10 a + 28\right)\cdot 29^{4} + 9\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 28 + 21\cdot 29 + 14\cdot 29^{2} + 21\cdot 29^{3} + 29^{4} + 8\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 26 a + 12 + \left(25 a + 16\right)\cdot 29 + \left(23 a + 9\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(18 a + 5\right)\cdot 29^{4} + \left(28 a + 1\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 9 a + 15 + \left(9 a + 7\right)\cdot 29 + \left(2 a + 26\right)\cdot 29^{2} + \left(4 a + 24\right)\cdot 29^{3} + \left(19 a + 22\right)\cdot 29^{4} + \left(27 a + 12\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 20 a + 2 + \left(19 a + 16\right)\cdot 29 + \left(26 a + 28\right)\cdot 29^{2} + \left(24 a + 13\right)\cdot 29^{3} + \left(9 a + 27\right)\cdot 29^{4} + \left(a + 15\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 20 + 4\cdot 29 + 20\cdot 29^{2} + 11\cdot 29^{3} + 26\cdot 29^{4} + 27\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 11 a + 9 + \left(6 a + 12\right)\cdot 29 + \left(18 a + 1\right)\cdot 29^{2} + \left(28 a + 21\right)\cdot 29^{3} + \left(15 a + 19\right)\cdot 29^{4} + \left(26 a + 19\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$

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.