Properties

Label 4.2e8_73.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{8} \cdot 73 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$18688= 2^{8} \cdot 73 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + x^{4} - x^{2} + 2 x + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.73.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 54 a + 81 + \left(10 a + 84\right)\cdot 89 + \left(25 a + 80\right)\cdot 89^{2} + \left(31 a + 68\right)\cdot 89^{3} + \left(53 a + 51\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 59 + 53\cdot 89 + 78\cdot 89^{2} + 63\cdot 89^{3} + 62\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 83 + 27\cdot 89 + 81\cdot 89^{2} + 33\cdot 89^{3} + 26\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 64 a + 59 + \left(62 a + 32\right)\cdot 89 + \left(34 a + 22\right)\cdot 89^{2} + \left(45 a + 44\right)\cdot 89^{3} + \left(80 a + 7\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 35 a + 14 + \left(78 a + 16\right)\cdot 89 + \left(63 a + 68\right)\cdot 89^{2} + \left(57 a + 84\right)\cdot 89^{3} + \left(35 a + 37\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 25 a + 62 + \left(26 a + 51\right)\cdot 89 + \left(54 a + 24\right)\cdot 89^{2} + \left(43 a + 60\right)\cdot 89^{3} + \left(8 a + 80\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$

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

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

Character values on conjugacy classes

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