Properties

Label 3.7e2_167.6t6.1c1
Dimension 3
Group $A_4\times C_2$
Conductor $ 7^{2} \cdot 167 $
Root number 1
Frobenius-Schur indicator 1

Related objects

Learn more about

Basic invariants

Dimension:$3$
Group:$A_4\times C_2$
Conductor:$8183= 7^{2} \cdot 167 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 5 x^{4} - 7 x^{3} + 10 x^{2} - 8 x + 8 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $A_4\times C_2$
Parity: Odd
Determinant: 1.167.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $ x^{2} + 24 x + 2 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 9 + \left(17 a + 9\right)\cdot 29 + \left(27 a + 4\right)\cdot 29^{2} + 25 a\cdot 29^{3} + \left(9 a + 18\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 + 20\cdot 29 + 26\cdot 29^{2} + 24\cdot 29^{3} + 25\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 18 a + 17 + \left(25 a + 1\right)\cdot 29 + \left(3 a + 11\right)\cdot 29^{2} + 12 a\cdot 29^{3} + \left(11 a + 22\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 11 a + 20 + \left(3 a + 24\right)\cdot 29 + \left(25 a + 4\right)\cdot 29^{2} + \left(16 a + 28\right)\cdot 29^{3} + \left(17 a + 8\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 15 + 26\cdot 29 + 29^{2} + 18\cdot 29^{3} + 28\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 26 a + 24 + \left(11 a + 4\right)\cdot 29 + \left(a + 9\right)\cdot 29^{2} + \left(3 a + 15\right)\cdot 29^{3} + \left(19 a + 12\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

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

Character values on conjugacy classes

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