Properties

Label 4.2e4_353.6t13.1c1
Dimension 4
Group $C_3^2:D_4$
Conductor $ 2^{4} \cdot 353 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$4$
Group:$C_3^2:D_4$
Conductor:$5648= 2^{4} \cdot 353 $
Artin number field: Splitting field of $f= x^{6} - 2 x^{5} + 2 x^{4} - x^{2} + 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $C_3^2:D_4$
Parity: Even
Determinant: 1.353.2t1.1c1

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:
$r_{ 1 }$ $=$ $ 2 + 23\cdot 29 + 28\cdot 29^{2} + 20\cdot 29^{3} + 22\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 15 a + 14 + \left(17 a + 9\right)\cdot 29 + 29^{2} + \left(22 a + 21\right)\cdot 29^{3} + \left(5 a + 19\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 18 a + 7 + \left(22 a + 14\right)\cdot 29 + \left(22 a + 20\right)\cdot 29^{2} + \left(10 a + 10\right)\cdot 29^{3} + \left(21 a + 23\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 14 a + 2 + \left(11 a + 24\right)\cdot 29 + \left(28 a + 15\right)\cdot 29^{2} + \left(6 a + 14\right)\cdot 29^{3} + \left(23 a + 26\right)\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 25 + 22\cdot 29 + 24\cdot 29^{2} + 6\cdot 29^{3} + 20\cdot 29^{4} +O\left(29^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 11 a + 10 + \left(6 a + 22\right)\cdot 29 + \left(6 a + 24\right)\cdot 29^{2} + \left(18 a + 12\right)\cdot 29^{3} + \left(7 a + 3\right)\cdot 29^{4} +O\left(29^{ 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.