Properties

Label 4.2e8_89.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{8} \cdot 89 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$22784= 2^{8} \cdot 89 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} - 3 x^{2} + 2 x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.89.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$: $ x^{2} + 70 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 26 a + 61 + \left(42 a + 9\right)\cdot 73 + 46 a\cdot 73^{2} + \left(51 a + 27\right)\cdot 73^{3} + \left(7 a + 64\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 4 + 55\cdot 73 + 22\cdot 73^{2} + 55\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 47 a + 66 + \left(30 a + 37\right)\cdot 73 + \left(26 a + 24\right)\cdot 73^{2} + \left(21 a + 62\right)\cdot 73^{3} + \left(65 a + 35\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 22 a + 18 + \left(25 a + 72\right)\cdot 73 + 73^{2} + \left(48 a + 42\right)\cdot 73^{3} + \left(39 a + 24\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 61 + 63\cdot 73 + 45\cdot 73^{2} + 47\cdot 73^{3} + 16\cdot 73^{4} +O\left(73^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 51 a + 11 + \left(47 a + 53\right)\cdot 73 + \left(72 a + 50\right)\cdot 73^{2} + \left(24 a + 39\right)\cdot 73^{3} + \left(33 a + 22\right)\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$4$
$6$$2$$(1,2)(3,4)(5,6)$$0$
$6$$2$$(3,5)$$2$
$9$$2$$(3,5)(4,6)$$0$
$4$$3$$(1,3,5)(2,4,6)$$-2$
$4$$3$$(1,3,5)$$1$
$18$$4$$(1,2)(3,6,5,4)$$0$
$12$$6$$(1,4,3,6,5,2)$$0$
$12$$6$$(2,4,6)(3,5)$$-1$
The blue line marks the conjugacy class containing complex conjugation.