Properties

Label 2.111.6t5.a.a
Dimension $2$
Group $S_3\times C_3$
Conductor $111$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: $S_3\times C_3$
Conductor: \(111\)\(\medspace = 3 \cdot 37 \)
Artin stem field: 6.0.36963.1
Galois orbit size: $2$
Smallest permutation container: $S_3\times C_3$
Parity: odd
Determinant: 1.111.6t1.b.b
Projective image: $S_3$
Projective stem field: 3.1.4107.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \(x^{2} + 7 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a + 2 + \left(2 a + 9\right)\cdot 11 + \left(10 a + 5\right)\cdot 11^{2} + \left(9 a + 2\right)\cdot 11^{3} + 3\cdot 11^{4} + \left(a + 10\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 6 + \left(5 a + 8\right)\cdot 11 + \left(4 a + 4\right)\cdot 11^{2} + \left(a + 2\right)\cdot 11^{3} + \left(8 a + 1\right)\cdot 11^{4} + \left(a + 3\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 3 }$ $=$ \( a + 8 + 5 a\cdot 11 + \left(4 a + 7\right)\cdot 11^{2} + 4 a\cdot 11^{3} + \left(4 a + 4\right)\cdot 11^{4} + \left(3 a + 8\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 4 a + 1 + \left(5 a + 2\right)\cdot 11 + \left(6 a + 6\right)\cdot 11^{2} + \left(9 a + 3\right)\cdot 11^{3} + \left(2 a + 10\right)\cdot 11^{4} + \left(9 a + 1\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 7 a + 7 + \left(8 a + 3\right)\cdot 11 + \left(a + 10\right)\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} + \left(9 a + 2\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 10 a + 1 + \left(5 a + 9\right)\cdot 11 + \left(6 a + 8\right)\cdot 11^{2} + \left(6 a + 2\right)\cdot 11^{3} + \left(6 a + 6\right)\cdot 11^{4} + \left(7 a + 6\right)\cdot 11^{5} +O(11^{6})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.