Properties

Label 3.154508.6t11.a.a
Dimension $3$
Group $S_4\times C_2$
Conductor $154508$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(154508\)\(\medspace = 2^{2} \cdot 19^{2} \cdot 107 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.618032.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: odd
Determinant: 1.107.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.870124.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 9.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.