Properties

Label 2.7_157.24t22.3c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 7 \cdot 157 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1099= 7 \cdot 157 $
Artin number field: Splitting field of $f= x^{8} - 6 x^{6} - 5 x^{5} + 19 x^{4} + 21 x^{3} - 18 x^{2} - 36 x - 5 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.7_157.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 1 + 27\cdot 29 + 25\cdot 29^{2} + 17\cdot 29^{3} + 17\cdot 29^{4} + 12\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 9 a + 23 + \left(13 a + 6\right)\cdot 29 + \left(20 a + 27\right)\cdot 29^{2} + \left(14 a + 8\right)\cdot 29^{3} + 24 a\cdot 29^{4} + 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 23 a + 20 + \left(24 a + 15\right)\cdot 29 + \left(24 a + 7\right)\cdot 29^{2} + \left(10 a + 25\right)\cdot 29^{3} + \left(17 a + 10\right)\cdot 29^{4} + \left(19 a + 24\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 6 a + 19 + 4 a\cdot 29 + \left(4 a + 20\right)\cdot 29^{2} + \left(18 a + 25\right)\cdot 29^{3} + \left(11 a + 28\right)\cdot 29^{4} + \left(9 a + 17\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 17 + 5\cdot 29 + 12\cdot 29^{2} + 14\cdot 29^{3} + 24\cdot 29^{4} + 23\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 11 a + \left(17 a + 18\right)\cdot 29 + \left(8 a + 27\right)\cdot 29^{2} + \left(16 a + 16\right)\cdot 29^{3} + \left(2 a + 22\right)\cdot 29^{4} + \left(8 a + 8\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 20 a + 10 + \left(15 a + 6\right)\cdot 29 + 8 a\cdot 29^{2} + \left(14 a + 4\right)\cdot 29^{3} + \left(4 a + 21\right)\cdot 29^{4} + \left(28 a + 9\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 18 a + 26 + \left(11 a + 6\right)\cdot 29 + \left(20 a + 24\right)\cdot 29^{2} + \left(12 a + 2\right)\cdot 29^{3} + \left(26 a + 19\right)\cdot 29^{4} + \left(20 a + 17\right)\cdot 29^{5} +O\left(29^{ 6 }\right)$

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

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

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)(2,6)(4,7)$$0$
$8$$3$$(1,4,3)(5,6,8)$$-1$
$6$$4$$(1,8,5,3)(2,6,7,4)$$0$
$8$$6$$(1,8,4,5,3,6)(2,7)$$1$
$6$$8$$(1,6,8,7,5,4,3,2)$$-\zeta_{8}^{3} - \zeta_{8}$
$6$$8$$(1,4,8,2,5,6,3,7)$$\zeta_{8}^{3} + \zeta_{8}$
The blue line marks the conjugacy class containing complex conjugation.