Properties

Label 3.749.6t11.b.a
Dimension $3$
Group $S_4\times C_2$
Conductor $749$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(749\)\(\medspace = 7 \cdot 107 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.80143.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: even
Determinant: 1.749.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.5243.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} + x^{4} + x^{3} + x^{2} - 2x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: \( x^{2} + 24x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 10 a + 26 + \left(15 a + 26\right)\cdot 29 + \left(3 a + 17\right)\cdot 29^{2} + \left(23 a + 18\right)\cdot 29^{3} + \left(12 a + 3\right)\cdot 29^{4} + \left(11 a + 1\right)\cdot 29^{5} + \left(20 a + 21\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 14 + 20\cdot 29 + 29^{3} + 16\cdot 29^{4} + 9\cdot 29^{5} + 23\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 12 a + 1 + \left(7 a + 16\right)\cdot 29 + 27\cdot 29^{2} + \left(22 a + 4\right)\cdot 29^{3} + \left(21 a + 6\right)\cdot 29^{4} + \left(2 a + 1\right)\cdot 29^{5} + \left(26 a + 21\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 a + 3 + \left(21 a + 12\right)\cdot 29 + \left(28 a + 21\right)\cdot 29^{2} + \left(6 a + 27\right)\cdot 29^{3} + \left(7 a + 5\right)\cdot 29^{4} + \left(26 a + 22\right)\cdot 29^{5} + \left(2 a + 3\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 27 + 4\cdot 29 + 28\cdot 29^{2} + 19\cdot 29^{3} + 10\cdot 29^{4} + 7\cdot 29^{5} + 22\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 19 a + 18 + \left(13 a + 6\right)\cdot 29 + \left(25 a + 20\right)\cdot 29^{2} + \left(5 a + 14\right)\cdot 29^{3} + \left(16 a + 15\right)\cdot 29^{4} + \left(17 a + 16\right)\cdot 29^{5} + \left(8 a + 24\right)\cdot 29^{6} +O(29^{7})\) Copy content Toggle raw display

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

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

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)$$1$
$3$$2$$(1,6)(3,4)$$-1$
$6$$2$$(2,3)(4,5)$$1$
$6$$2$$(1,6)(2,3)(4,5)$$-1$
$8$$3$$(1,3,2)(4,5,6)$$0$
$6$$4$$(1,4,6,3)$$1$
$6$$4$$(1,6)(2,4,5,3)$$-1$
$8$$6$$(1,4,5,6,3,2)$$0$

The blue line marks the conjugacy class containing complex conjugation.