Properties

Label 3.2e2_19e2.6t8.1c1
Dimension 3
Group $S_4$
Conductor $ 2^{2} \cdot 19^{2}$
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$1444= 2^{2} \cdot 19^{2} $
Artin number field: Splitting field of $f=x^{4} - x^{3} - 2 x^{2} - 6 x - 2$ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Even
Determinant: 1.1.1t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $x^{2} + 12 x + 2$
Roots: \[ \begin{aligned} r_{ 1 } &= 153500 a - 67100 +O\left(13^{ 5 }\right) \\ r_{ 2 } &= 37201 +O\left(13^{ 5 }\right) \\ r_{ 3 } &= 82535 +O\left(13^{ 5 }\right) \\ r_{ 4 } &= -153500 a - 52635 +O\left(13^{ 5 }\right) \\ \end{aligned}\]

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

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

Character values on conjugacy classes

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