Properties

Label 3.332.6t11.a.a
Dimension $3$
Group $S_4\times C_2$
Conductor $332$
Root number $1$
Indicator $1$

Related objects

Downloads

Learn more

Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(332\)\(\medspace = 2^{2} \cdot 83 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.27556.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: even
Determinant: 1.332.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.1328.1

Defining polynomial

$f(x)$$=$ \( x^{6} - x^{5} + 2x^{3} - x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 20 a + 21 + \left(19 a + 9\right)\cdot 41 + \left(16 a + 18\right)\cdot 41^{2} + 13\cdot 41^{3} + \left(33 a + 3\right)\cdot 41^{4} + \left(15 a + 23\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 25 a + 19 + \left(33 a + 33\right)\cdot 41 + \left(3 a + 26\right)\cdot 41^{2} + \left(17 a + 34\right)\cdot 41^{3} + \left(13 a + 34\right)\cdot 41^{4} + \left(25 a + 39\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 19 + 18\cdot 41^{2} + 15\cdot 41^{3} + 39\cdot 41^{4} + 35\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 13 + 4\cdot 41 + 7\cdot 41^{2} + 20\cdot 41^{3} + 8\cdot 41^{4} + 7\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 16 a + 12 + \left(7 a + 27\right)\cdot 41 + \left(37 a + 4\right)\cdot 41^{2} + 23 a\cdot 41^{3} + \left(27 a + 17\right)\cdot 41^{4} + \left(15 a + 20\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 21 a + 40 + \left(21 a + 6\right)\cdot 41 + \left(24 a + 7\right)\cdot 41^{2} + \left(40 a + 39\right)\cdot 41^{3} + \left(7 a + 19\right)\cdot 41^{4} + \left(25 a + 37\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,3)(4,6)$
$(1,2,3)(4,6,5)$
$(3,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$3$
$1$$2$$(1,6)(2,5)(3,4)$$-3$
$3$$2$$(1,6)(3,4)$$-1$
$3$$2$$(1,6)$$1$
$6$$2$$(1,3)(4,6)$$1$
$6$$2$$(1,6)(2,3)(4,5)$$-1$
$8$$3$$(1,2,3)(4,6,5)$$0$
$6$$4$$(1,4,6,3)$$1$
$6$$4$$(1,4,6,3)(2,5)$$-1$
$8$$6$$(1,4,5,6,3,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.