Properties

Label 2.3e2_239.24t22.1c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 3^{2} \cdot 239 $
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$2151= 3^{2} \cdot 239 $
Artin number field: Splitting field of $f= x^{8} - 3 x^{7} - 8 x^{6} + 21 x^{5} + 21 x^{4} - 33 x^{3} - 14 x^{2} + 39 x - 8 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.239.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 38 a + 20 + \left(42 a + 5\right)\cdot 43 + \left(16 a + 23\right)\cdot 43^{2} + \left(31 a + 8\right)\cdot 43^{3} + \left(42 a + 7\right)\cdot 43^{4} + \left(36 a + 3\right)\cdot 43^{5} + 16\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 2 }$ $=$ $ 41 + 21\cdot 43 + 3\cdot 43^{2} + 25\cdot 43^{3} + 42\cdot 43^{4} + 31\cdot 43^{5} + 17\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 3 }$ $=$ $ 12 + 8\cdot 43 + 5\cdot 43^{2} + 15\cdot 43^{3} + 7\cdot 43^{4} + 8\cdot 43^{5} + 4\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 4 }$ $=$ $ 36 a + 38 + \left(20 a + 36\right)\cdot 43 + \left(33 a + 39\right)\cdot 43^{2} + \left(18 a + 25\right)\cdot 43^{3} + \left(20 a + 35\right)\cdot 43^{4} + \left(9 a + 37\right)\cdot 43^{5} + \left(38 a + 25\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 5 }$ $=$ $ 7 a + 27 + \left(a + 36\right)\cdot 43 + \left(8 a + 21\right)\cdot 43^{2} + \left(4 a + 33\right)\cdot 43^{3} + \left(20 a + 3\right)\cdot 43^{4} + \left(36 a + 25\right)\cdot 43^{5} + \left(6 a + 8\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 6 }$ $=$ $ 5 a + 15 + 10\cdot 43 + \left(26 a + 40\right)\cdot 43^{2} + \left(11 a + 22\right)\cdot 43^{3} + 18\cdot 43^{4} + \left(6 a + 40\right)\cdot 43^{5} + \left(42 a + 22\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 7 }$ $=$ $ 7 a + 31 + \left(22 a + 21\right)\cdot 43 + \left(9 a + 9\right)\cdot 43^{2} + \left(24 a + 11\right)\cdot 43^{3} + \left(22 a + 37\right)\cdot 43^{4} + \left(33 a + 26\right)\cdot 43^{5} + \left(4 a + 11\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$
$r_{ 8 }$ $=$ $ 36 a + 34 + \left(41 a + 30\right)\cdot 43 + \left(34 a + 28\right)\cdot 43^{2} + \left(38 a + 29\right)\cdot 43^{3} + \left(22 a + 19\right)\cdot 43^{4} + \left(6 a + 41\right)\cdot 43^{5} + \left(36 a + 21\right)\cdot 43^{6} +O\left(43^{ 7 }\right)$

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

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

Character values on conjugacy classes

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