Properties

Label 2.2243.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2243 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2243 $
Artin number field: Splitting field of $f= x^{8} - 6 x^{6} - 9 x^{5} + 11 x^{4} + 35 x^{3} + 4 x^{2} - 74 x - 85 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.2243.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 1 + 29\cdot 41 + 33\cdot 41^{2} + 23\cdot 41^{3} + 13\cdot 41^{4} +O\left(41^{ 6 }\right)$
$r_{ 2 }$ $=$ $ 37 a + 38 + \left(24 a + 29\right)\cdot 41 + \left(33 a + 27\right)\cdot 41^{2} + \left(38 a + 16\right)\cdot 41^{3} + \left(8 a + 24\right)\cdot 41^{4} + \left(35 a + 9\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 3 }$ $=$ $ 4 a + 31 + \left(33 a + 33\right)\cdot 41 + \left(2 a + 12\right)\cdot 41^{2} + \left(24 a + 25\right)\cdot 41^{3} + \left(30 a + 36\right)\cdot 41^{4} + \left(14 a + 12\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 4 }$ $=$ $ 24 + 31\cdot 41 + 2\cdot 41^{2} + 34\cdot 41^{3} + 40\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 5 }$ $=$ $ 37 a + 2 + \left(7 a + 6\right)\cdot 41 + \left(38 a + 29\right)\cdot 41^{2} + \left(16 a + 12\right)\cdot 41^{3} + \left(10 a + 22\right)\cdot 41^{4} + \left(26 a + 26\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 6 }$ $=$ $ 4 a + 26 + \left(16 a + 26\right)\cdot 41 + \left(7 a + 21\right)\cdot 41^{2} + \left(2 a + 17\right)\cdot 41^{3} + \left(32 a + 12\right)\cdot 41^{4} + \left(5 a + 24\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 7 }$ $=$ $ 20 a + 32 + \left(15 a + 10\right)\cdot 41 + \left(32 a + 18\right)\cdot 41^{2} + \left(15 a + 9\right)\cdot 41^{3} + \left(39 a + 37\right)\cdot 41^{4} + \left(14 a + 1\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$
$r_{ 8 }$ $=$ $ 21 a + 10 + \left(25 a + 37\right)\cdot 41 + \left(8 a + 17\right)\cdot 41^{2} + \left(25 a + 24\right)\cdot 41^{3} + \left(a + 16\right)\cdot 41^{4} + \left(26 a + 7\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$

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

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

Character values on conjugacy classes

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