Properties

Label 2.3e2_239.24t22.2c2
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 3^{2} \cdot 239 $
Root number not computed
Frobenius-Schur indicator 0

Related objects

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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} - 4 x^{7} + 4 x^{6} + 2 x^{5} - 8 x^{4} + 8 x^{3} + 10 x^{2} - 13 x - 5 $ 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 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: $ x^{2} + 42 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 29 a + 24 + \left(38 a + 22\right)\cdot 43 + \left(33 a + 31\right)\cdot 43^{2} + \left(20 a + 41\right)\cdot 43^{3} + \left(3 a + 19\right)\cdot 43^{4} + \left(42 a + 22\right)\cdot 43^{5} + \left(38 a + 41\right)\cdot 43^{6} + \left(17 a + 38\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$
$r_{ 2 }$ $=$ $ 9 + 34\cdot 43 + 23\cdot 43^{2} + 16\cdot 43^{3} + 16\cdot 43^{5} + 3\cdot 43^{6} + 12\cdot 43^{7} +O\left(43^{ 8 }\right)$
$r_{ 3 }$ $=$ $ 35 + 8\cdot 43 + 19\cdot 43^{2} + 26\cdot 43^{3} + 42\cdot 43^{4} + 26\cdot 43^{5} + 39\cdot 43^{6} + 30\cdot 43^{7} +O\left(43^{ 8 }\right)$
$r_{ 4 }$ $=$ $ 22 a + 11 + \left(18 a + 23\right)\cdot 43 + \left(26 a + 17\right)\cdot 43^{2} + \left(23 a + 1\right)\cdot 43^{3} + \left(7 a + 8\right)\cdot 43^{4} + \left(23 a + 35\right)\cdot 43^{5} + \left(11 a + 5\right)\cdot 43^{6} + \left(40 a + 7\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$
$r_{ 5 }$ $=$ $ 21 a + 33 + \left(24 a + 19\right)\cdot 43 + \left(16 a + 25\right)\cdot 43^{2} + \left(19 a + 41\right)\cdot 43^{3} + \left(35 a + 34\right)\cdot 43^{4} + \left(19 a + 7\right)\cdot 43^{5} + \left(31 a + 37\right)\cdot 43^{6} + \left(2 a + 35\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$
$r_{ 6 }$ $=$ $ 14 a + 10 + \left(4 a + 32\right)\cdot 43 + \left(9 a + 26\right)\cdot 43^{2} + \left(22 a + 28\right)\cdot 43^{3} + \left(39 a + 2\right)\cdot 43^{4} + 18\cdot 43^{5} + \left(4 a + 38\right)\cdot 43^{6} + \left(25 a + 17\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$
$r_{ 7 }$ $=$ $ 29 a + 34 + \left(38 a + 10\right)\cdot 43 + \left(33 a + 16\right)\cdot 43^{2} + \left(20 a + 14\right)\cdot 43^{3} + \left(3 a + 40\right)\cdot 43^{4} + \left(42 a + 24\right)\cdot 43^{5} + \left(38 a + 4\right)\cdot 43^{6} + \left(17 a + 25\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$
$r_{ 8 }$ $=$ $ 14 a + 20 + \left(4 a + 20\right)\cdot 43 + \left(9 a + 11\right)\cdot 43^{2} + \left(22 a + 1\right)\cdot 43^{3} + \left(39 a + 23\right)\cdot 43^{4} + 20\cdot 43^{5} + \left(4 a + 1\right)\cdot 43^{6} + \left(25 a + 4\right)\cdot 43^{7} +O\left(43^{ 8 }\right)$

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

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

Character values on conjugacy classes

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