Properties

Label 3.23232.6t11.a.a
Dimension $3$
Group $S_4\times C_2$
Conductor $23232$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $3$
Group: $S_4\times C_2$
Conductor: \(23232\)\(\medspace = 2^{6} \cdot 3 \cdot 11^{2} \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.2811072.1
Galois orbit size: $1$
Smallest permutation container: $S_4\times C_2$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_4$
Projective stem field: Galois closure of 4.2.17424.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} - x^{4} + 2x^{3} + 2x^{2} - 8x + 14 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2\cdot 7 + 6\cdot 7^{2} + 2\cdot 7^{3} + 5\cdot 7^{4} + 2\cdot 7^{5} + 4\cdot 7^{6} + 4\cdot 7^{7} + 6\cdot 7^{8} + 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( a + 4 + \left(6 a + 3\right)\cdot 7 + \left(2 a + 6\right)\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{3} + \left(a + 5\right)\cdot 7^{4} + 4 a\cdot 7^{5} + 6 a\cdot 7^{6} + \left(6 a + 3\right)\cdot 7^{7} + \left(5 a + 1\right)\cdot 7^{8} + \left(6 a + 1\right)\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 + 7 + 6\cdot 7^{2} + 2\cdot 7^{4} + 6\cdot 7^{5} + 3\cdot 7^{6} + 4\cdot 7^{7} + 3\cdot 7^{8} + 3\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 5 + 7 + \left(4 a + 3\right)\cdot 7^{2} + \left(a + 5\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} + \left(2 a + 3\right)\cdot 7^{5} + 2\cdot 7^{6} + 3\cdot 7^{7} + a\cdot 7^{8} + 2\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + \left(4 a + 6\right)\cdot 7 + 4\cdot 7^{2} + \left(2 a + 3\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} + \left(4 a + 4\right)\cdot 7^{5} + \left(5 a + 4\right)\cdot 7^{6} + 2 a\cdot 7^{7} + \left(5 a + 3\right)\cdot 7^{8} + \left(6 a + 5\right)\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 3 a + 4 + \left(2 a + 6\right)\cdot 7 + 6 a\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(a + 4\right)\cdot 7^{4} + \left(2 a + 3\right)\cdot 7^{5} + \left(a + 5\right)\cdot 7^{6} + \left(4 a + 4\right)\cdot 7^{7} + \left(a + 5\right)\cdot 7^{8} + 6\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.