Properties

Label 3.1399.4t5.1c1
Dimension 3
Group $S_4$
Conductor $ 1399 $
Root number 1
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$1399 $
Artin number field: Splitting field of $f= x^{4} - x^{3} + x - 2 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Odd
Determinant: 1.1399.2t1.1c1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
$r_{ 1 }$ $=$ $ 12 + 13\cdot 37 + 7\cdot 37^{2} + 2\cdot 37^{3} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 18 + 27\cdot 37 + 35\cdot 37^{3} + 27\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 22 + 29\cdot 37 + 11\cdot 37^{2} + 26\cdot 37^{3} + 25\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 23 + 3\cdot 37 + 17\cdot 37^{2} + 10\cdot 37^{3} + 20\cdot 37^{4} +O\left(37^{ 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.