Properties

Label 3.2e3_149.4t5.1c1
Dimension 3
Group $S_4$
Conductor $ 2^{3} \cdot 149 $
Root number 1
Frobenius-Schur indicator 1

Related objects

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$1192= 2^{3} \cdot 149 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + 2 x^{2} + x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Odd
Determinant: 1.2e3_149.2t1.2c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 167 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 82 + 52\cdot 167 + 4\cdot 167^{2} + 112\cdot 167^{3} + 98\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 130 + 59\cdot 167 + 30\cdot 167^{2} + 141\cdot 167^{3} + 118\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 133 + 132\cdot 167 + 55\cdot 167^{2} + 33\cdot 167^{3} + 8\cdot 167^{4} +O\left(167^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 157 + 88\cdot 167 + 76\cdot 167^{2} + 47\cdot 167^{3} + 108\cdot 167^{4} +O\left(167^{ 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.