Properties

Label 3.7344.4t5.a.a
Dimension 3
Group $S_4$
Conductor $ 2^{4} \cdot 3^{3} \cdot 17 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$7344= 2^{4} \cdot 3^{3} \cdot 17 $
Artin number field: Splitting field of 4.2.7344.1 defined by $f= x^{4} - 2 x^{3} - 2 x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Odd
Determinant: 1.51.2t1.a.a
Projective image: $S_4$
Projective field: Galois closure of 4.2.7344.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 151 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 21 + 60\cdot 151 + 137\cdot 151^{2} + 90\cdot 151^{3} + 81\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 51 + 35\cdot 151 + 116\cdot 151^{2} + 40\cdot 151^{3} +O\left(151^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 89 + 137\cdot 151 + 135\cdot 151^{2} + 90\cdot 151^{3} + 104\cdot 151^{4} +O\left(151^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 143 + 68\cdot 151 + 63\cdot 151^{2} + 79\cdot 151^{3} + 115\cdot 151^{4} +O\left(151^{ 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.