Properties

Label 2.491.24t22.1c2
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:
$r_{ 1 }$ $=$ $ 37 a + 31 + \left(34 a + 5\right)\cdot 47 + \left(4 a + 25\right)\cdot 47^{2} + \left(27 a + 24\right)\cdot 47^{3} + \left(33 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 5 a + 20 + \left(46 a + 32\right)\cdot 47 + \left(39 a + 37\right)\cdot 47^{2} + \left(6 a + 34\right)\cdot 47^{3} + \left(7 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 10 a + 11 + \left(12 a + 38\right)\cdot 47 + \left(42 a + 46\right)\cdot 47^{2} + \left(19 a + 26\right)\cdot 47^{3} + \left(13 a + 35\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 42 a + 30 + 25\cdot 47 + \left(7 a + 24\right)\cdot 47^{2} + \left(40 a + 8\right)\cdot 47^{3} + \left(39 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 8 a + 44 + \left(19 a + 21\right)\cdot 47 + \left(21 a + 17\right)\cdot 47^{2} + \left(15 a + 45\right)\cdot 47^{3} + \left(3 a + 5\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 35 + 19\cdot 47 + 34\cdot 47^{2} + 23\cdot 47^{3} + 39\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 7 }$ $=$ $ 39 a + 13 + \left(27 a + 5\right)\cdot 47 + \left(25 a + 41\right)\cdot 47^{2} + \left(31 a + 7\right)\cdot 47^{3} + \left(43 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 8 }$ $=$ $ 6 + 39\cdot 47 + 7\cdot 47^{2} + 16\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$

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.