Properties

Label 2.2e3_43.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 43 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

Learn more about

Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$344= 2^{3} \cdot 43 $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 6 x^{3} - 2 x^{2} - 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.43.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{2} + 33 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 35 a + 33 + \left(17 a + 1\right)\cdot 37 + \left(10 a + 15\right)\cdot 37^{2} + \left(28 a + 24\right)\cdot 37^{3} + \left(36 a + 8\right)\cdot 37^{4} + \left(6 a + 19\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 27 a + 22 + \left(24 a + 34\right)\cdot 37 + \left(27 a + 4\right)\cdot 37^{2} + \left(25 a + 36\right)\cdot 37^{3} + \left(28 a + 19\right)\cdot 37^{4} + \left(11 a + 1\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 13 a + 7 + \left(33 a + 19\right)\cdot 37 + \left(10 a + 31\right)\cdot 37^{2} + \left(14 a + 1\right)\cdot 37^{3} + \left(15 a + 13\right)\cdot 37^{4} + \left(13 a + 2\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 27 + 14\cdot 37 + 5\cdot 37^{2} + 15\cdot 37^{3} + 13\cdot 37^{4} + 35\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 34 + 14\cdot 37 + 30\cdot 37^{2} + 5\cdot 37^{3} + 18\cdot 37^{4} + 18\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 10 a + 19 + \left(12 a + 32\right)\cdot 37 + \left(9 a + 16\right)\cdot 37^{2} + 11 a\cdot 37^{3} + \left(8 a + 35\right)\cdot 37^{4} + \left(25 a + 19\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 24 a + 22 + \left(3 a + 28\right)\cdot 37 + \left(26 a + 4\right)\cdot 37^{2} + \left(22 a + 11\right)\cdot 37^{3} + \left(21 a + 23\right)\cdot 37^{4} + \left(23 a + 3\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 2 a + 25 + \left(19 a + 1\right)\cdot 37 + \left(26 a + 2\right)\cdot 37^{2} + \left(8 a + 16\right)\cdot 37^{3} + 16\cdot 37^{4} + \left(30 a + 10\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$

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

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

Character values on conjugacy classes

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