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:
$r_{ 1 }$ $=$ $ 9 a + 6 + \left(3 a + 12\right)\cdot 13 + \left(11 a + 5\right)\cdot 13^{2} + \left(4 a + 8\right)\cdot 13^{3} + \left(5 a + 10\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 8 + 13 + 12\cdot 13^{2} + 3\cdot 13^{3} + 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 11 + 4\cdot 13 + 7\cdot 13^{2} + 11\cdot 13^{3} + 2\cdot 13^{4} +O\left(13^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 4 a + 2 + \left(9 a + 7\right)\cdot 13 + a\cdot 13^{2} + \left(8 a + 2\right)\cdot 13^{3} + \left(7 a + 11\right)\cdot 13^{4} +O\left(13^{ 5 }\right)$

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.