Properties

Label 2.3e3_41.24t22.3c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 3^{3} \cdot 41 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1107= 3^{3} \cdot 41 $
Artin number field: Splitting field of $f= x^{8} - x^{7} + x^{6} + 5 x^{5} - 11 x^{4} + 8 x^{3} - 7 x^{2} + 4 x - 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.3_41.2t1.1c1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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