Properties

Label 2.3e2_173.24t22.1c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 3^{2} \cdot 173 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1557= 3^{2} \cdot 173 $
Artin number field: Splitting field of $f= x^{8} - 2 x^{7} - 2 x^{6} + x^{5} + 4 x^{4} + 10 x^{3} - 11 x^{2} - 14 x - 20 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.3_173.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 8 a + 5 + \left(15 a + 4\right)\cdot 23 + \left(22 a + 16\right)\cdot 23^{2} + \left(13 a + 18\right)\cdot 23^{3} + \left(16 a + 18\right)\cdot 23^{4} + \left(13 a + 11\right)\cdot 23^{5} + 11 a\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 14 + \left(3 a + 22\right)\cdot 23 + \left(6 a + 3\right)\cdot 23^{2} + \left(19 a + 20\right)\cdot 23^{3} + \left(9 a + 19\right)\cdot 23^{4} + \left(16 a + 22\right)\cdot 23^{5} + \left(6 a + 11\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 15 a + 21 + \left(7 a + 3\right)\cdot 23 + \left(9 a + 1\right)\cdot 23^{3} + \left(6 a + 15\right)\cdot 23^{4} + \left(9 a + 22\right)\cdot 23^{5} + \left(11 a + 9\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 2 a + 6 + \left(9 a + 3\right)\cdot 23 + \left(2 a + 7\right)\cdot 23^{2} + \left(21 a + 4\right)\cdot 23^{3} + \left(5 a + 3\right)\cdot 23^{4} + \left(6 a + 3\right)\cdot 23^{5} + \left(9 a + 10\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 20 a + 20 + \left(19 a + 2\right)\cdot 23 + \left(16 a + 13\right)\cdot 23^{2} + \left(3 a + 6\right)\cdot 23^{3} + \left(13 a + 20\right)\cdot 23^{4} + \left(6 a + 22\right)\cdot 23^{5} + \left(16 a + 8\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 22 + 18\cdot 23 + 14\cdot 23^{2} + 20\cdot 23^{3} + 14\cdot 23^{4} + 15\cdot 23^{5} + 14\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 19 + 16\cdot 23 + 10\cdot 23^{2} + 22\cdot 23^{3} + 5\cdot 23^{4} + 6\cdot 23^{5} + 13\cdot 23^{6} +O\left(23^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 21 a + 10 + \left(13 a + 19\right)\cdot 23 + \left(20 a + 2\right)\cdot 23^{2} + \left(a + 21\right)\cdot 23^{3} + \left(17 a + 16\right)\cdot 23^{4} + \left(16 a + 9\right)\cdot 23^{5} + \left(13 a + 22\right)\cdot 23^{6} +O\left(23^{ 7 }\right)$

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

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

Character values on conjugacy classes

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