Properties

Label 2.283.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 283 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$283 $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} + 4 x^{6} - 8 x^{5} + 8 x^{4} - x^{3} + 2 x^{2} - 3 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.283.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $ x^{2} + 16 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 2 a + 1 + \left(6 a + 12\right)\cdot 17 + \left(8 a + 11\right)\cdot 17^{2} + \left(7 a + 5\right)\cdot 17^{3} + \left(9 a + 6\right)\cdot 17^{4} + \left(10 a + 11\right)\cdot 17^{5} + 11\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 15 a + 15 + \left(2 a + 6\right)\cdot 17 + \left(a + 2\right)\cdot 17^{2} + \left(16 a + 6\right)\cdot 17^{3} + \left(12 a + 4\right)\cdot 17^{4} + \left(3 a + 12\right)\cdot 17^{5} + \left(a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 15 a + 3 + \left(10 a + 16\right)\cdot 17 + \left(8 a + 13\right)\cdot 17^{2} + \left(9 a + 4\right)\cdot 17^{3} + \left(7 a + 8\right)\cdot 17^{4} + \left(6 a + 12\right)\cdot 17^{5} + \left(16 a + 1\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 14 + 14\cdot 17^{3} + 11\cdot 17^{4} + 9\cdot 17^{5} + 3\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 3 + \left(3 a + 10\right)\cdot 17 + \left(2 a + 9\right)\cdot 17^{2} + \left(7 a + 2\right)\cdot 17^{3} + \left(11 a + 5\right)\cdot 17^{4} + \left(14 a + 9\right)\cdot 17^{5} + \left(10 a + 7\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 12 + 3\cdot 17 + 4\cdot 17^{2} + 6\cdot 17^{3} + 4\cdot 17^{4} + 14\cdot 17^{5} + 4\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 2 a + 13 + \left(14 a + 11\right)\cdot 17 + 15 a\cdot 17^{2} + 4\cdot 17^{3} + \left(4 a + 1\right)\cdot 17^{4} + \left(13 a + 3\right)\cdot 17^{5} + \left(15 a + 16\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 10 a + 10 + \left(13 a + 6\right)\cdot 17 + \left(14 a + 8\right)\cdot 17^{2} + \left(9 a + 7\right)\cdot 17^{3} + \left(5 a + 9\right)\cdot 17^{4} + \left(2 a + 12\right)\cdot 17^{5} + \left(6 a + 3\right)\cdot 17^{6} +O\left(17^{ 7 }\right)$

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

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

Character values on conjugacy classes

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