Properties

Label 4.2e6_97.6t13.2c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{6} \cdot 97 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$6208= 2^{6} \cdot 97 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.97.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
$r_{ 1 }$ $=$ $ 12 + 44\cdot 47 + 40\cdot 47^{2} + 2\cdot 47^{3} + 14\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 3 a + 35 + \left(41 a + 3\right)\cdot 47 + \left(21 a + 27\right)\cdot 47^{2} + \left(5 a + 46\right)\cdot 47^{3} + \left(27 a + 40\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 10 + 19\cdot 47 + 13\cdot 47^{2} + 11\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 44 a + 41 + \left(5 a + 35\right)\cdot 47 + \left(25 a + 29\right)\cdot 47^{2} + \left(41 a + 35\right)\cdot 47^{3} + \left(19 a + 42\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 17 a + 29 + \left(18 a + 32\right)\cdot 47 + \left(3 a + 20\right)\cdot 47^{2} + \left(34 a + 13\right)\cdot 47^{3} + \left(13 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 30 a + 16 + \left(28 a + 5\right)\cdot 47 + \left(43 a + 9\right)\cdot 47^{2} + \left(12 a + 31\right)\cdot 47^{3} + \left(33 a + 13\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

Character values on conjugacy classes

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