Properties

Label 2.2e3_163.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 163 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1304= 2^{3} \cdot 163 $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 10 x^{6} - 14 x^{5} + 4 x^{4} + 14 x^{3} + 6 x^{2} - 8 x - 10 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.163.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 67 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 67 }$: $ x^{2} + 63 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 26 a + 10 + \left(59 a + 40\right)\cdot 67 + 15\cdot 67^{2} + \left(16 a + 24\right)\cdot 67^{3} + \left(14 a + 12\right)\cdot 67^{4} + 17\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 9 + 18\cdot 67 + 50\cdot 67^{2} + 58\cdot 67^{3} + 28\cdot 67^{4} + 38\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 41 a + 47 + \left(7 a + 50\right)\cdot 67 + \left(66 a + 26\right)\cdot 67^{2} + \left(50 a + 20\right)\cdot 67^{3} + \left(52 a + 53\right)\cdot 67^{4} + \left(66 a + 3\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 44 a + 19 + \left(60 a + 47\right)\cdot 67 + \left(41 a + 2\right)\cdot 67^{2} + \left(20 a + 66\right)\cdot 67^{3} + 46\cdot 67^{4} + \left(20 a + 9\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 23 a + 61 + \left(6 a + 44\right)\cdot 67 + \left(25 a + 42\right)\cdot 67^{2} + \left(46 a + 39\right)\cdot 67^{3} + \left(66 a + 27\right)\cdot 67^{4} + \left(46 a + 22\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 7 a + 5 + \left(33 a + 36\right)\cdot 67 + \left(18 a + 43\right)\cdot 67^{2} + \left(11 a + 2\right)\cdot 67^{3} + \left(33 a + 59\right)\cdot 67^{4} + \left(61 a + 47\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 60 a + 33 + \left(33 a + 27\right)\cdot 67 + \left(48 a + 17\right)\cdot 67^{2} + \left(55 a + 29\right)\cdot 67^{3} + \left(33 a + 46\right)\cdot 67^{4} + \left(5 a + 59\right)\cdot 67^{5} +O\left(67^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 21 + 3\cdot 67 + 2\cdot 67^{2} + 27\cdot 67^{3} + 60\cdot 67^{4} + 67^{5} +O\left(67^{ 6 }\right)$

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

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

Character values on conjugacy classes

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