Properties

Label 3.11_233.4t5.1
Dimension 3
Group $S_4$
Conductor $ 11 \cdot 233 $
Frobenius-Schur indicator 1

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Basic invariants

Dimension:$3$
Group:$S_4$
Conductor:$2563= 11 \cdot 233 $
Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 4 x^{2} - x - 1 $ over $\Q$
Size of Galois orbit: 1
Smallest containing permutation representation: $S_4$
Parity: Odd

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $ x^{2} + 6 x + 3 $
Roots:
$r_{ 1 }$ $=$ $ 3 a + 1 + \left(3 a + 2\right)\cdot 7 + \left(6 a + 5\right)\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + \left(3 a + 1\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 6 + 2\cdot 7 + 2\cdot 7^{2} + 6\cdot 7^{3} + 6\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 4 a + 4 + \left(3 a + 2\right)\cdot 7 + 7^{2} + \left(4 a + 2\right)\cdot 7^{3} + \left(3 a + 2\right)\cdot 7^{4} +O\left(7^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 5 + 6\cdot 7 + 4\cdot 7^{2} + 6\cdot 7^{3} + 2\cdot 7^{4} +O\left(7^{ 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 values
$c1$
$1$ $1$ $()$ $3$
$3$ $2$ $(1,2)(3,4)$ $-1$
$6$ $2$ $(1,2)$ $1$
$8$ $3$ $(1,2,3)$ $0$
$6$ $4$ $(1,2,3,4)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.