Properties

Label 2.2e2_523.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{2} \cdot 523 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2092= 2^{2} \cdot 523 $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 3 x^{6} - x^{5} + 11 x^{4} - 9 x^{3} + 2 x^{2} - 9 x - 2 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.523.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 10 a + 29 + \left(17 a + 30\right)\cdot 37 + \left(33 a + 21\right)\cdot 37^{2} + \left(26 a + 34\right)\cdot 37^{3} + \left(31 a + 17\right)\cdot 37^{4} + \left(25 a + 7\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 30 + 34\cdot 37 + 28\cdot 37^{2} + 5\cdot 37^{3} + 11\cdot 37^{4} + 14\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 27 a + 32 + \left(19 a + 15\right)\cdot 37 + \left(3 a + 27\right)\cdot 37^{2} + \left(10 a + 34\right)\cdot 37^{3} + \left(5 a + 6\right)\cdot 37^{4} + \left(11 a + 5\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 12 a + 22 + 18\cdot 37 + \left(21 a + 4\right)\cdot 37^{2} + \left(10 a + 7\right)\cdot 37^{3} + 34 a\cdot 37^{4} + \left(21 a + 29\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 20 a + 30 + \left(21 a + 24\right)\cdot 37 + \left(27 a + 35\right)\cdot 37^{2} + \left(22 a + 36\right)\cdot 37^{3} + \left(6 a + 8\right)\cdot 37^{4} + \left(14 a + 24\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 17 a + 36 + \left(15 a + 16\right)\cdot 37 + \left(9 a + 13\right)\cdot 37^{2} + \left(14 a + 26\right)\cdot 37^{3} + \left(30 a + 12\right)\cdot 37^{4} + 22 a\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 25 a + 33 + \left(36 a + 7\right)\cdot 37 + \left(15 a + 14\right)\cdot 37^{2} + \left(26 a + 28\right)\cdot 37^{3} + \left(2 a + 15\right)\cdot 37^{4} + \left(15 a + 8\right)\cdot 37^{5} +O\left(37^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 14 + 35\cdot 37 + 37^{2} + 11\cdot 37^{3} + 22\cdot 37^{5} +O\left(37^{ 6 }\right)$

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

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

Character values on conjugacy classes

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