Properties

Label 2.2e3_3e2_5e2.24t22.3c1
Dimension 2
Group $\textrm{GL(2,3)}$
Conductor $ 2^{3} \cdot 3^{2} \cdot 5^{2}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$2$
Group:$\textrm{GL(2,3)}$
Conductor:$1800= 2^{3} \cdot 3^{2} \cdot 5^{2} $
Artin number field: Splitting field of $f= x^{8} - 4 x^{7} + 4 x^{6} - 16 x^{5} + 10 x^{4} - 16 x^{3} + 4 x^{2} - 4 x + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: 24T22
Parity: Odd
Determinant: 1.3.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
$r_{ 1 }$ $=$ $ 29 a + \left(8 a + 12\right)\cdot 41 + \left(a + 7\right)\cdot 41^{2} + \left(5 a + 31\right)\cdot 41^{3} + \left(25 a + 11\right)\cdot 41^{4} + \left(22 a + 37\right)\cdot 41^{5} + \left(27 a + 23\right)\cdot 41^{6} + \left(23 a + 38\right)\cdot 41^{7} + \left(30 a + 25\right)\cdot 41^{8} + \left(40 a + 20\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$
$r_{ 2 }$ $=$ $ 12 a + 5 + \left(32 a + 9\right)\cdot 41 + \left(39 a + 2\right)\cdot 41^{2} + \left(35 a + 4\right)\cdot 41^{3} + 15 a\cdot 41^{4} + \left(18 a + 39\right)\cdot 41^{5} + \left(13 a + 1\right)\cdot 41^{6} + 17 a\cdot 41^{7} + \left(10 a + 12\right)\cdot 41^{8} + 30\cdot 41^{9} +O\left(41^{ 10 }\right)$
$r_{ 3 }$ $=$ $ 26 a + 9 + \left(6 a + 27\right)\cdot 41 + \left(30 a + 17\right)\cdot 41^{2} + \left(27 a + 14\right)\cdot 41^{3} + \left(34 a + 18\right)\cdot 41^{4} + \left(40 a + 4\right)\cdot 41^{5} + \left(11 a + 30\right)\cdot 41^{6} + \left(28 a + 1\right)\cdot 41^{7} + \left(34 a + 10\right)\cdot 41^{8} + \left(33 a + 28\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$
$r_{ 4 }$ $=$ $ 33 + 14\cdot 41 + 12\cdot 41^{2} + 6\cdot 41^{3} + 29\cdot 41^{4} + 26\cdot 41^{5} + 28\cdot 41^{6} + 34\cdot 41^{7} + 24\cdot 41^{8} + 34\cdot 41^{9} +O\left(41^{ 10 }\right)$
$r_{ 5 }$ $=$ $ 5 + 40\cdot 41 + 10\cdot 41^{2} + 33\cdot 41^{3} + 34\cdot 41^{4} + 28\cdot 41^{5} + 21\cdot 41^{6} + 38\cdot 41^{7} + 20\cdot 41^{8} + 10\cdot 41^{9} +O\left(41^{ 10 }\right)$
$r_{ 6 }$ $=$ $ 15 a + 5 + \left(34 a + 21\right)\cdot 41 + \left(10 a + 19\right)\cdot 41^{2} + \left(13 a + 26\right)\cdot 41^{3} + \left(6 a + 12\right)\cdot 41^{4} + 10\cdot 41^{5} + \left(29 a + 25\right)\cdot 41^{6} + \left(12 a + 33\right)\cdot 41^{7} + \left(6 a + 3\right)\cdot 41^{8} + \left(7 a + 13\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$
$r_{ 7 }$ $=$ $ 37 a + \left(22 a + 4\right)\cdot 41 + 37\cdot 41^{2} + \left(9 a + 10\right)\cdot 41^{3} + \left(20 a + 23\right)\cdot 41^{4} + \left(35 a + 6\right)\cdot 41^{5} + \left(38 a + 37\right)\cdot 41^{6} + \left(30 a + 1\right)\cdot 41^{7} + \left(12 a + 9\right)\cdot 41^{8} + \left(26 a + 21\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$
$r_{ 8 }$ $=$ $ 4 a + 29 + \left(18 a + 35\right)\cdot 41 + \left(40 a + 15\right)\cdot 41^{2} + \left(31 a + 37\right)\cdot 41^{3} + \left(20 a + 33\right)\cdot 41^{4} + \left(5 a + 10\right)\cdot 41^{5} + \left(2 a + 36\right)\cdot 41^{6} + \left(10 a + 14\right)\cdot 41^{7} + \left(28 a + 16\right)\cdot 41^{8} + \left(14 a + 5\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$

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

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

Character values on conjugacy classes

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