Properties

Label 3.2e2_5_443.6t11.2c1
Dimension 3
Group $S_4\times C_2$
Conductor $ 2^{2} \cdot 5 \cdot 443 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4\times C_2$
Conductor:$8860= 2^{2} \cdot 5 \cdot 443 $
Artin number field: Splitting field of $f=x^{6} - 2 x^{5} - 5 x^{4} + 6 x^{3} + 22 x^{2} - 8 x - 32$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: 6T11
Parity: Even
Determinant: 1.2e2_5_443.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{2} + 38 x + 6$
Roots: \[ \begin{aligned} r_{ 1 } &= 32284446522069 +O\left(41^{ 9 }\right) \\ r_{ 2 } &= -17018632151894 a - 96648293081573 +O\left(41^{ 9 }\right) \\ r_{ 3 } &= -99146499116417 +O\left(41^{ 9 }\right) \\ r_{ 4 } &= -144207146833476 a + 13226928202367 +O\left(41^{ 9 }\right) \\ r_{ 5 } &= 17018632151894 a - 104704140097523 +O\left(41^{ 9 }\right) \\ r_{ 6 } &= 144207146833476 a - 72394376822882 +O\left(41^{ 9 }\right) \\ \end{aligned}\]

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$1$$2$$(1,3)(2,5)(4,6)$$-3$
$3$$2$$(1,3)$$1$
$3$$2$$(1,3)(2,5)$$-1$
$6$$2$$(2,4)(5,6)$$1$
$6$$2$$(1,3)(2,4)(5,6)$$-1$
$8$$3$$(1,2,4)(3,5,6)$$0$
$6$$4$$(1,5,3,2)$$1$
$6$$4$$(1,3)(2,6,5,4)$$-1$
$8$$6$$(1,5,6,3,2,4)$$0$
The blue line marks the conjugacy class containing complex conjugation.