Properties

Label 3.727207.6t11.b.a
Dimension $3$
Group $S_4\times C_2$
Conductor $727207$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(727207\)\(\medspace = 13^{3} \cdot 331 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.240705517.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: odd
Determinant: 1.4303.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.331.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} - 15x^{4} + 42x^{3} + 51x^{2} + 13x - 182 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \( x^{2} + 21x + 5 \) Copy content Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.